Pub Date : 2025-12-01DOI: 10.1016/j.dam.2025.11.046
Yulin Chang , Jichang Wu , Zhiwei Zhang
We study the existence of directed Hamilton cycles and oriented spanning trees of bounded degree in randomly perturbed sparse digraphs. Let be a digraph on vertices with minimum semi-degree , where , and let be the binomial random digraph on vertices in which each of possible ordered pairs forms an arc independently with probability . We prove that, for any , if , then contains a directed Hamilton cycle with high probability. Moreover, if , and , then is vertex-pancyclic with high probability. Finally, we also prove a similar result for oriented spanning trees in this model.
研究了随机摄动稀疏有向图中有向Hamilton环和有界度有向生成树的存在性。设Dα是n个顶点上最小半度δ0(Dα)≥αn的有向图，其中0<；α<1；设D（n,p）是n个顶点上的二项随机有向图，其中n（n−1）个可能的有序对中的每一个都以概率p独立形成一个弧。我们证明了对于任意α=α(n): n =(0,1)，若β=(6+o(1))log1α，则Dα∪D（n,β/n）包含一个高概率的有向Hamilton环。如果α=ω(n−1/8),β=β(n)=6log1α，则Dα∪D（n,β/n）是高概率顶点泛环。最后，我们也证明了该模型中有向生成树的类似结果。
{"title":"Cycles and trees in randomly perturbed sparse digraphs","authors":"Yulin Chang , Jichang Wu , Zhiwei Zhang","doi":"10.1016/j.dam.2025.11.046","DOIUrl":"10.1016/j.dam.2025.11.046","url":null,"abstract":"<div><div>We study the existence of directed Hamilton cycles and oriented spanning trees of bounded degree in randomly perturbed sparse digraphs. Let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> be a digraph on <span><math><mi>n</mi></math></span> vertices with minimum semi-degree <span><math><mrow><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>)</mo></mrow><mo>≥</mo><mi>α</mi><mi>n</mi></mrow></math></span>, where <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></math></span>, and let <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> be the binomial random digraph on <span><math><mi>n</mi></math></span> vertices in which each of <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> possible ordered pairs forms an arc independently with probability <span><math><mi>p</mi></math></span>. We prove that, for any <span><math><mrow><mi>α</mi><mo>=</mo><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mi>N</mi><mo>↦</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, if <span><math><mrow><mi>β</mi><mo>=</mo><mrow><mo>(</mo><mn>6</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>log</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi></mrow></mfrac></mrow></math></span>, then <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>∪</mo><mi>D</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>β</mi><mo>/</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> contains a directed Hamilton cycle with high probability. Moreover, if <span><math><mrow><mi>α</mi><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>8</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><mi>β</mi><mo>=</mo><mi>β</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mo>log</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi></mrow></mfrac></mrow></math></span>, then <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>∪</mo><mi>D</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>β</mi><mo>/</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is vertex-pancyclic with high probability. Finally, we also prove a similar result for oriented spanning trees in this model.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 176-182"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.dam.2025.11.050
Lin Li , Rong-Xia Hao , Yan-Quan Feng , Jaeun Lee , Eddie Cheng
<div><div>Let <span><math><mi>G</mi></math></span> be a graph with vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Two trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> of <span><math><mi>G</mi></math></span> are <em>internally disjoint</em> <span><math><mi>S</mi></math></span><em>-tree</em> for <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mo>0̸</mo></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></math></span>. The <span><math><mi>r</mi></math></span><em>-tree connectivity</em> <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is <span><math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>r</mi><mo>}</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum number of internally pairwise disjoint <span><math><mi>S</mi></math></span>-trees in <span><math><mi>G</mi></math></span>. The lower bound of <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> for Cartesian product graph <span><math><mrow><mi>G</mi><mo>□</mo><mi>H</mi></mrow></math></span> of connected graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> is characterized in Gao et al. (2018). In this paper, we obtained a lower bound of <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>4</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> which depends only on the <span><math><mi>k</mi></math></span>-tree connectivity of <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> with <span><math><mrow><mi>k</mi><mo>≤</mo><mn>4</mn></mrow></math></span>. Fur
{"title":"The lower bounds of 4-tree connectivity of Cartesian product graphs","authors":"Lin Li , Rong-Xia Hao , Yan-Quan Feng , Jaeun Lee , Eddie Cheng","doi":"10.1016/j.dam.2025.11.050","DOIUrl":"10.1016/j.dam.2025.11.050","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Two trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> of <span><math><mi>G</mi></math></span> are <em>internally disjoint</em> <span><math><mi>S</mi></math></span><em>-tree</em> for <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mo>0̸</mo></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></math></span>. The <span><math><mi>r</mi></math></span><em>-tree connectivity</em> <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is <span><math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>r</mi><mo>}</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum number of internally pairwise disjoint <span><math><mi>S</mi></math></span>-trees in <span><math><mi>G</mi></math></span>. The lower bound of <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> for Cartesian product graph <span><math><mrow><mi>G</mi><mo>□</mo><mi>H</mi></mrow></math></span> of connected graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> is characterized in Gao et al. (2018). In this paper, we obtained a lower bound of <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>4</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> which depends only on the <span><math><mi>k</mi></math></span>-tree connectivity of <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> with <span><math><mrow><mi>k</mi><mo>≤</mo><mn>4</mn></mrow></math></span>. Fur","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 155-165"},"PeriodicalIF":1.0,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-28DOI: 10.1016/j.dam.2025.11.048
Zemin Jin, Huifang Liu, Qing Jie
<div><div>In this paper, we consider two extremal problems about 4-cycles in multipartite graphs. Denote by <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup></math></span> the 4-cycle in a multipartite graph whose vertices come from exactly four different partite sets. We call a 4-cycle in a multipartite graph <em>multipartite</em>, denoted by <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>m</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi></mrow></msubsup></math></span>, if its vertices come from at least three different partite sets. An edge-colored graph is called <em>rainbow</em> if any two edges of it have different colors. For given graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the anti-Ramsey number <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of colors in an edge-colored <span><math><mi>G</mi></math></span> with no rainbow <span><math><mi>H</mi></math></span>. This graph parameter is closely related to the Turán number. These two parameters on 3-cycles in general complete multipartite graphs have been well determined. The anti-Ramsey number on 4-cycles was solved in general complete <span><math><mi>r</mi></math></span>-partite graphs, while the number on multipartite 4-cycle was only determined for <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span>. The Turán number on (multipartite) 4-cycles in complete <span><math><mi>r</mi></math></span>-partite graphs was proved only for <span><math><mrow><mi>r</mi><mo>≤</mo><mn>3</mn></mrow></math></span>. In this paper, we show that <span><math><mrow><mi>e</mi><mi>x</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><m
{"title":"Two extremal problems for 4-cycles in 4-partite graphs","authors":"Zemin Jin, Huifang Liu, Qing Jie","doi":"10.1016/j.dam.2025.11.048","DOIUrl":"10.1016/j.dam.2025.11.048","url":null,"abstract":"<div><div>In this paper, we consider two extremal problems about 4-cycles in multipartite graphs. Denote by <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup></math></span> the 4-cycle in a multipartite graph whose vertices come from exactly four different partite sets. We call a 4-cycle in a multipartite graph <em>multipartite</em>, denoted by <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>m</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi></mrow></msubsup></math></span>, if its vertices come from at least three different partite sets. An edge-colored graph is called <em>rainbow</em> if any two edges of it have different colors. For given graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the anti-Ramsey number <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of colors in an edge-colored <span><math><mi>G</mi></math></span> with no rainbow <span><math><mi>H</mi></math></span>. This graph parameter is closely related to the Turán number. These two parameters on 3-cycles in general complete multipartite graphs have been well determined. The anti-Ramsey number on 4-cycles was solved in general complete <span><math><mi>r</mi></math></span>-partite graphs, while the number on multipartite 4-cycle was only determined for <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span>. The Turán number on (multipartite) 4-cycles in complete <span><math><mi>r</mi></math></span>-partite graphs was proved only for <span><math><mrow><mi>r</mi><mo>≤</mo><mn>3</mn></mrow></math></span>. In this paper, we show that <span><math><mrow><mi>e</mi><mi>x</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 166-175"},"PeriodicalIF":1.0,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-27DOI: 10.1016/j.dam.2025.11.037
Christian Komusiewicz , Jannik Schestag
In the Generalized Noah’s Ark Problem, one is given a phylogenetic tree on a set of species and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of Generalized Noah’s Ark Problem and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species, .
{"title":"A multivariate complexity analysis of the Generalized Noah’s Ark Problem","authors":"Christian Komusiewicz , Jannik Schestag","doi":"10.1016/j.dam.2025.11.037","DOIUrl":"10.1016/j.dam.2025.11.037","url":null,"abstract":"<div><div>In the <span>Generalized Noah’s Ark Problem</span>, one is given a phylogenetic tree on a set of species <span><math><mi>X</mi></math></span> and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of <span>Generalized Noah’s Ark Problem</span> and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species, <span><math><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 137-154"},"PeriodicalIF":1.0,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The -irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph as , where and denote the degrees of vertices and , respectively. Recent research has successfully characterized chemical trees with the maximum -irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree . Application of these properties enables us to characterize maximal trees with We establish that extremal trees contain only vertices of degrees 2 and . Moreover, the number of edges with both end-vertices having the degree 2 or is very small, so almost all edges have the (second) maximum possible contribution to -irregularity. We believe this property or similar should extend to maximal trees for any value of , so this is an interesting direction for further research.
{"title":"The σ-irregularity of trees with maximum degree 5","authors":"Darko Dimitrov , Žana Kovijanić Vukićević , Goran Popivoda , Jelena Sedlar , Riste Škrekovski , Saša Vujošević","doi":"10.1016/j.dam.2025.11.045","DOIUrl":"10.1016/j.dam.2025.11.045","url":null,"abstract":"<div><div>The <span><math><mi>σ</mi></math></span>-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> as <span><math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denote the degrees of vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span>, respectively. Recent research has successfully characterized chemical trees with the maximum <span><math><mi>σ</mi></math></span>-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree <span><math><mi>Δ</mi></math></span>. Application of these properties enables us to characterize maximal trees with <span><math><mrow><mi>Δ</mi><mo>=</mo><mn>5</mn><mo>.</mo></mrow></math></span> We establish that extremal trees contain only vertices of degrees <span><math><mrow><mn>1</mn><mo>,</mo></mrow></math></span> 2 and <span><math><mi>Δ</mi></math></span>. Moreover, the number of edges with both end-vertices having the degree 2 or <span><math><mi>Δ</mi></math></span> is very small, so almost all edges have the (second) maximum possible contribution to <span><math><mi>σ</mi></math></span>-irregularity. We believe this property or similar should extend to maximal trees for any value of <span><math><mi>Δ</mi></math></span>, so this is an interesting direction for further research.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 124-136"},"PeriodicalIF":1.0,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.dam.2025.11.034
Yanqing Wu
A -injective-edge coloring of a graph is an edge coloring such that any two edges at distance 2 or in the same triangle are assigned different colors. The injective chromatic index of a graph , denoted by , is the smallest integer for which has a -injective-edge coloring. Additionally, the maximum average degree of a graph , denoted by , is defined as the largest average degree among all subgraphs of . In this paper, we prove that if is a graph with maximum degree 5 and , then . This result improves upon the previous bound for a graph with maximum degree 5 and established by Zhu et al. (2024).
图G的k-内射边着色是c:E(G)→{1,2，…，k}的边着色，使得距离为2的任意两条边或同一三角形中的任意两条边被赋予不同的颜色。图G的内射着色指数，用χi ' (G)表示，是使G具有k-内射边着色的最小整数k。另外，图G的最大平均度用mad(G)定义为图G的所有子图H (G)中最大的平均度。本文证明了如果G是最大度为5且mad(G)<；72的图，则χi ' (G)≤17。该结果改进了之前由Zhu et al.（2024）建立的最大度为5和mad(G)<；72的图G的界χi ' (G)≤19。
{"title":"On injective edge coloring for a class of graphs with maximum degree 5","authors":"Yanqing Wu","doi":"10.1016/j.dam.2025.11.034","DOIUrl":"10.1016/j.dam.2025.11.034","url":null,"abstract":"<div><div>A <span><math><mi>k</mi></math></span>-injective-edge coloring of a graph <span><math><mi>G</mi></math></span> is an edge coloring <span><math><mrow><mi>c</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> such that any two edges at distance 2 or in the same triangle are assigned different colors. The injective chromatic index of a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the smallest integer <span><math><mi>k</mi></math></span> for which <span><math><mi>G</mi></math></span> has a <span><math><mi>k</mi></math></span>-injective-edge coloring. Additionally, the maximum average degree of a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mtext>mad</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is defined as the largest average degree among all subgraphs <span><math><mi>H</mi></math></span> of <span><math><mi>G</mi></math></span>. In this paper, we prove that if <span><math><mi>G</mi></math></span> is a graph with maximum degree 5 and <span><math><mrow><mtext>mad</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo><</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, then <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>17</mn></mrow></math></span>. This result improves upon the previous bound <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>19</mn></mrow></math></span> for a graph <span><math><mi>G</mi></math></span> with maximum degree 5 and <span><math><mrow><mtext>mad</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo><</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> established by Zhu et al. (2024).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 313-320"},"PeriodicalIF":1.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.dam.2025.11.040
Károly Bezdek , Zsolt Lángi , Márton Naszódi
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the -dimensional real vector space, mostly equipped with the Euclidean norm. Our first topic is the Kneser–Poulsen Conjecture, according to which if a finite number of balls are rearranged so that the pairwise distances of the centers increase, then the volume of the union (resp., intersection) increases (resp., decreases).
Next, we discuss Blaschke–Santaló-type inequalities, and reverse isoperimetric inequalities for convex sets in Euclidean -space obtained as intersections of (possibly infinitely many) balls of radius , which we call -ball bodies. We present some results on 1-ball bodies (also called ball-bodies or spindle convex sets) in the plane, with special attention paid to their approximation by the spindle convex hull of a finite subset.
A ball-polyhedron is a ball-body obtained as the intersection of finitely many unit balls in Euclidean -space. We consider the combinatorial structure of their faces, and volumetric properties of ball-polyhedra obtained from choosing the centers of the balls randomly.
{"title":"Selected topics from the theory of intersections of balls","authors":"Károly Bezdek , Zsolt Lángi , Márton Naszódi","doi":"10.1016/j.dam.2025.11.040","DOIUrl":"10.1016/j.dam.2025.11.040","url":null,"abstract":"<div><div>In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the <span><math><mi>d</mi></math></span>-dimensional real vector space, mostly equipped with the Euclidean norm. Our first topic is the <em>Kneser–Poulsen Conjecture</em>, according to which if a finite number of balls are rearranged so that the pairwise distances of the centers increase, then the volume of the union (resp., intersection) increases (resp., decreases).</div><div>Next, we discuss Blaschke–Santaló-type inequalities, and reverse isoperimetric inequalities for convex sets in Euclidean <span><math><mi>d</mi></math></span>-space obtained as intersections of (possibly infinitely many) balls of radius <span><math><mi>r</mi></math></span>, which we call <span><math><mi>r</mi></math></span>-<em>ball bodies</em>. We present some results on 1-ball bodies (also called <em>ball-bodies</em> or <em>spindle convex sets</em>) in the plane, with special attention paid to their approximation by the spindle convex hull of a finite subset.</div><div>A <em>ball-polyhedron</em> is a ball-body obtained as the intersection of finitely many unit balls in Euclidean <span><math><mi>d</mi></math></span>-space. We consider the combinatorial structure of their faces, and volumetric properties of ball-polyhedra obtained from choosing the centers of the balls randomly.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 60-82"},"PeriodicalIF":1.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.dam.2025.11.013
Ararat Harutyunyan , Hovhannes A. Harutyunyan , Aram Khanlari
Given a graph , the broadcast time of is the least number of time units required to disseminate one piece of information starting at an arbitrary originating vertex in , where each transmission takes one time unit and each informed vertex can transmit the message to one of its neighbors at any time. We initiate the study of finding upper bounds on broadcast time in highly connected graphs. In particular, we give upper bounds on for -connected graphs and graphs with a large minimum degree. We prove that for all -connected graphs. For many families of graphs this bound is tight. We also show that if the minimum degree of is at least , then , and derive various similar upper bounds on graphs with large minimum degree. Finally, we discuss an open problem that relates the broadcast time of a graph with its minimum degree, when the latter is small.
{"title":"Bounds on broadcast time in well-connected graphs","authors":"Ararat Harutyunyan , Hovhannes A. Harutyunyan , Aram Khanlari","doi":"10.1016/j.dam.2025.11.013","DOIUrl":"10.1016/j.dam.2025.11.013","url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi></math></span>, the <em>broadcast time</em> <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is the least number of time units required to disseminate one piece of information starting at an arbitrary originating vertex in <span><math><mi>G</mi></math></span>, where each transmission takes one time unit and each informed vertex can transmit the message to one of its neighbors at any time. We initiate the study of finding upper bounds on broadcast time <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> in highly connected graphs. In particular, we give upper bounds on <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mi>k</mi></math></span>-connected graphs and graphs with a large minimum degree. We prove that <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌈</mo><mo>log</mo><mi>k</mi><mo>⌉</mo></mrow><mo>+</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> for all <span><math><mi>k</mi></math></span>-connected graphs. For many families of graphs this bound is tight. We also show that if the minimum degree of <span><math><mi>G</mi></math></span> is at least <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span>, then <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌈</mo><mo>log</mo><mi>n</mi><mo>⌉</mo></mrow><mo>+</mo><mn>3</mn></mrow></math></span>, and derive various similar upper bounds on graphs with large minimum degree. Finally, we discuss an open problem that relates the broadcast time of a graph with its minimum degree, when the latter is small.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 83-90"},"PeriodicalIF":1.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.dam.2025.11.019
Jean-Claude Bermond, Michel Cosnard
The Maximum All Request Path Grooming (MARPG) problem consists in finding the maximum number of requests (connections) which can be established in a network, where each arc has a given capacity or bandwidth (grooming factor). The Maximum Path Coloring problem consists for a given number of colors (wavelengths) in finding the maximum number of requests that can be established so that two requests sharing an arc have different colors. These problems are part of the more general RWA (Routing and Wavelength Assignment) problem and have been studied for various classes of networks like paths, dipaths, undirected trees and symmetric directed trees. Here we consider the case where the network is an oriented tree (tree in which each edge has a unique orientation) where the two problems are equivalent. We give the value of the maximum number of requests for various families of oriented trees like Fig-Trees. To do that we revisit the problem when the network is a directed path by giving the structure of a maximum set of requests and determining bounds on the maximum load of an arc of the dipath. These bounds can be used for computing the cutwidth of a graph.
{"title":"Maximizing the number of requests in oriented trees with a grooming factor","authors":"Jean-Claude Bermond, Michel Cosnard","doi":"10.1016/j.dam.2025.11.019","DOIUrl":"10.1016/j.dam.2025.11.019","url":null,"abstract":"<div><div>The Maximum All Request Path Grooming (MARPG) problem consists in finding the maximum number of requests (connections) which can be established in a network, where each arc has a given capacity or bandwidth <span><math><mi>C</mi></math></span> (grooming factor). The Maximum Path Coloring problem consists for a given number of colors (wavelengths) <span><math><mi>W</mi></math></span> in finding the maximum number of requests that can be established so that two requests sharing an arc have different colors. These problems are part of the more general RWA (Routing and Wavelength Assignment) problem and have been studied for various classes of networks like paths, dipaths, undirected trees and symmetric directed trees. Here we consider the case where the network is an oriented tree (tree in which each edge has a unique orientation) where the two problems are equivalent. We give the value of the maximum number of requests for various families of oriented trees like Fig-Trees. To do that we revisit the problem when the network is a directed path by giving the structure of a maximum set of requests and determining bounds on the maximum load of an arc of the dipath. These bounds can be used for computing the cutwidth of a graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 91-107"},"PeriodicalIF":1.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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