Pub Date : 2026-01-29DOI: 10.1016/j.disc.2026.115024
Jing Yu , Jie-Xiang Zhu
We study the enumeration of graph orientations under local degree constraints. Given a finite graph and a family of admissible sets , let denote the number of orientations in which the out-degree of each vertex v lies in . We prove a general duality formula expressing as a signed sum over edge subsets, involving products of coefficient sums associated with , from a family of polynomials. Our approach employs gauge transformations, a technique rooted in statistical physics and holographic algorithms. We also present a probabilistic derivation of the same identity, interpreting the orientation-generating polynomial as the expectation of a random polynomial product. As applications, we obtain explicit formulas for the number of even orientations and for mixed Eulerian-even orientations on general graphs. Our formula generalizes a result of Borbényi and Csikvári on Eulerian orientations of graphs.
{"title":"Counting degree-constrained orientations","authors":"Jing Yu , Jie-Xiang Zhu","doi":"10.1016/j.disc.2026.115024","DOIUrl":"10.1016/j.disc.2026.115024","url":null,"abstract":"<div><div>We study the enumeration of graph orientations under local degree constraints. Given a finite graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> and a family of admissible sets <span><math><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>⊆</mo><mi>Z</mi><mo>:</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>}</mo></math></span>, let <span><math><mi>N</mi><mo>(</mo><mi>G</mi><mo>;</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math></span> denote the number of orientations in which the out-degree of each vertex <em>v</em> lies in <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span>. We prove a general duality formula expressing <span><math><mi>N</mi><mo>(</mo><mi>G</mi><mo>;</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math></span> as a signed sum over edge subsets, involving products of coefficient sums associated with <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub></math></span>, from a family of polynomials. Our approach employs gauge transformations, a technique rooted in statistical physics and holographic algorithms. We also present a probabilistic derivation of the same identity, interpreting the orientation-generating polynomial as the expectation of a random polynomial product. As applications, we obtain explicit formulas for the number of even orientations and for mixed Eulerian-even orientations on general graphs. Our formula generalizes a result of Borbényi and Csikvári on Eulerian orientations of graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115024"},"PeriodicalIF":0.7,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079007","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 : 2026-01-29DOI: 10.1016/j.disc.2026.115019
Zixuan Yang , Hongliang Lu , Shenggui Zhang
In this paper, we determine the maximum number of edges of connected graphs with given maximum degree and matching number. This gives an answer to a problem posed by Dibek et al. (2017) [6]. We also show that the bound in our result is tight.
{"title":"Maximum size of connected graphs with bounded maximum degree and matching number","authors":"Zixuan Yang , Hongliang Lu , Shenggui Zhang","doi":"10.1016/j.disc.2026.115019","DOIUrl":"10.1016/j.disc.2026.115019","url":null,"abstract":"<div><div>In this paper, we determine the maximum number of edges of connected graphs with given maximum degree and matching number. This gives an answer to a problem posed by Dibek et al. (2017) <span><span>[6]</span></span>. We also show that the bound in our result is tight.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 7","pages":"Article 115019"},"PeriodicalIF":0.7,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081593","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 : 2026-01-29DOI: 10.1016/j.disc.2026.115021
Harry Sapranidis Mantelos
<div><div>We study the problem raised in Marco Stevens (2021) <span><span>[20]</span></span> concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples.</div><div>The problem is plainly stated as follows: Let Λ be a set and <span><math><mi>F</mi></math></span> a field, and suppose that <span><math><mi>K</mi><mo>,</mo><mi>Q</mi><mo>:</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>F</mi></math></span> are two functions such that for any <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi>Λ</mi></math></span>, the determinants of matrices <span><math><msub><mrow><mo>(</mo><mi>K</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>(</mo><mi>Q</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> agree. What are all the possible transformations that transform <em>Q</em> into <em>K</em>? In Marco Stevens (2021) <span><span>[20]</span></span> the following two were conjectured: <span><math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>; and <span><math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>g</mi><msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> for some nowhere-zero function <em>g</em>. In the same paper, this conjectured classification is verified in the case of symmetric functions <em>K</em> and <em>Q</em>. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed.</div><div>By taking Λ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23–64]. Hence, o
{"title":"Determinantally equivalent nonzero functions","authors":"Harry Sapranidis Mantelos","doi":"10.1016/j.disc.2026.115021","DOIUrl":"10.1016/j.disc.2026.115021","url":null,"abstract":"<div><div>We study the problem raised in Marco Stevens (2021) <span><span>[20]</span></span> concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples.</div><div>The problem is plainly stated as follows: Let Λ be a set and <span><math><mi>F</mi></math></span> a field, and suppose that <span><math><mi>K</mi><mo>,</mo><mi>Q</mi><mo>:</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>F</mi></math></span> are two functions such that for any <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi>Λ</mi></math></span>, the determinants of matrices <span><math><msub><mrow><mo>(</mo><mi>K</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mo>(</mo><mi>Q</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> agree. What are all the possible transformations that transform <em>Q</em> into <em>K</em>? In Marco Stevens (2021) <span><span>[20]</span></span> the following two were conjectured: <span><math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>; and <span><math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>g</mi><msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> for some nowhere-zero function <em>g</em>. In the same paper, this conjectured classification is verified in the case of symmetric functions <em>K</em> and <em>Q</em>. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed.</div><div>By taking Λ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23–64]. Hence, o","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115021"},"PeriodicalIF":0.7,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078349","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 : 2026-01-28DOI: 10.1016/j.disc.2026.115003
Masaki Kashima , Riste Škrekovski , Rongxing Xu
A proper coloring ϕ of G is called a proper conflict-free coloring of G if for every non-isolated vertex v of G, there is a color c such that . As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free -choosability of graphs. For a non-negative integer k, a graph G is proper conflict-free -choosable if for any list assignment L of G with for every vertex , G admits a proper conflict-free coloring ϕ such that for every vertex . In this note, we first remark if a graph G is d-degenerate, then G is proper conflict-free -choosable. Furthermore, when , we can reduce the number of colors by showing that every tree is proper conflict-free -choosable. This motivates us to state a question.
{"title":"Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy","authors":"Masaki Kashima , Riste Škrekovski , Rongxing Xu","doi":"10.1016/j.disc.2026.115003","DOIUrl":"10.1016/j.disc.2026.115003","url":null,"abstract":"<div><div>A proper coloring <em>ϕ</em> of <em>G</em> is called a proper conflict-free coloring of <em>G</em> if for every non-isolated vertex <em>v</em> of <em>G</em>, there is a color <em>c</em> such that <span><math><mo>|</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>∩</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosability of graphs. For a non-negative integer <em>k</em>, a graph <em>G</em> is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosable if for any list assignment <em>L</em> of <em>G</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>+</mo><mi>k</mi></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>G</em> admits a proper conflict-free coloring <em>ϕ</em> such that <span><math><mi>ϕ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>∈</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this note, we first remark if a graph <em>G</em> is <em>d</em>-degenerate, then <em>G</em> is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable. Furthermore, when <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, we can reduce the number of colors by showing that every tree is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable. This motivates us to state a question.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115003"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079006","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 : 2026-01-28DOI: 10.1016/j.disc.2026.115017
L. Sunil Chandran , Jinia Ghosh
The boxicity (respectively, cubicity) of an undirected graph Γ is the smallest non-negative integer k such that Γ can be represented as the intersection graph of axis-parallel rectangular boxes (respectively, unit cubes) in . An undirected graph is classified as a comparability graph if it is isomorphic to the comparability graph of some partial order. In this paper, we initiate the study of boxicity and cubicity for two subclasses of comparability graphs - divisor graphs and power graphs.
Divisor graphs, an important family of comparability graphs, arise from a number-theoretically defined poset, namely the divisibility poset. We consider one of the most popular subclasses of divisor graphs, denoted by , where the vertex set is the set of positive divisors of a natural number n, and two vertices a and b are adjacent if and only if a divides b or b divides a. We derive estimates, tight up to a factor of 2, for the boxicity and cubicity of .
Power graphs are a special class of algebraically defined comparability graphs. The power graph of a group is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We show that studying the boxicity (respectively, cubicity) of is sufficient to study the boxicity (respectively, cubicity) of the power graph of the cyclic group of order n. Thus, as a corollary of our first result, we derive similar estimates for the boxicity and cubicity power graphs of cyclic groups.
{"title":"Boxicity and cubicity of a subclass of divisor graphs and power graphs of cyclic groups","authors":"L. Sunil Chandran , Jinia Ghosh","doi":"10.1016/j.disc.2026.115017","DOIUrl":"10.1016/j.disc.2026.115017","url":null,"abstract":"<div><div>The <em>boxicity</em> (respectively, <em>cubicity</em>) of an undirected graph Γ is the smallest non-negative integer <em>k</em> such that Γ can be represented as the intersection graph of axis-parallel rectangular boxes (respectively, unit cubes) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>. An undirected graph is classified as a <em>comparability graph</em> if it is isomorphic to the comparability graph of some partial order. In this paper, we initiate the study of boxicity and cubicity for two subclasses of comparability graphs - <em>divisor graphs</em> and <em>power graphs</em>.</div><div>Divisor graphs, an important family of comparability graphs, arise from a number-theoretically defined poset, namely the <em>divisibility poset</em>. We consider one of the most popular subclasses of divisor graphs, denoted by <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where the vertex set is the set of positive divisors of a natural number <em>n</em>, and two vertices <em>a</em> and <em>b</em> are adjacent if and only if <em>a</em> divides <em>b</em> or <em>b</em> divides <em>a</em>. We derive estimates, tight up to a factor of 2, for the boxicity and cubicity of <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div><div>Power graphs are a special class of algebraically defined comparability graphs. The power graph of a group is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We show that studying the boxicity (respectively, cubicity) of <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is sufficient to study the boxicity (respectively, cubicity) of the power graph of the cyclic group of order <em>n</em>. Thus, as a corollary of our first result, we derive similar estimates for the boxicity and cubicity power graphs of cyclic groups.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 7","pages":"Article 115017"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049173","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 : 2026-01-28DOI: 10.1016/j.disc.2026.115001
Andrew Lane , Natasha Morrison
Given a graph H, we say that a graph G is properly rainbow H-saturated if: (1) There is a proper edge colouring of G containing no rainbow copy of H; (2) For every , every proper edge colouring of contains a rainbow copy of H. The proper rainbow saturation number is the minimum number of edges in a properly rainbow H-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on for general cliques, cycles, and complete bipartite graphs. We also provide a necessary and sufficient condition for a graph to have constant proper rainbow saturation number.
{"title":"Improved bounds for proper rainbow saturation","authors":"Andrew Lane , Natasha Morrison","doi":"10.1016/j.disc.2026.115001","DOIUrl":"10.1016/j.disc.2026.115001","url":null,"abstract":"<div><div>Given a graph <em>H</em>, we say that a graph <em>G</em> is <em>properly rainbow H-saturated</em> if: (1) There is a proper edge colouring of <em>G</em> containing no rainbow copy of <em>H</em>; (2) For every <span><math><mi>e</mi><mo>∉</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, every proper edge colouring of <span><math><mi>G</mi><mo>+</mo><mi>e</mi></math></span> contains a rainbow copy of <em>H</em>. The <em>proper rainbow saturation number</em> <span><math><mrow><mi>sa</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the minimum number of edges in a properly rainbow <em>H</em>-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on <span><math><mrow><mi>sa</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for general cliques, cycles, and complete bipartite graphs. We also provide a necessary and sufficient condition for a graph to have constant proper rainbow saturation number.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115001"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079004","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 : 2026-01-28DOI: 10.1016/j.disc.2026.115018
Steven T. Dougherty , Esengül Saltürk
We describe the neighbor construction for self-dual codes over and give the type of the neighbor based on the type of the code and vector v used to construct the neighbor. We define the neighbor graph of self-dual codes over as the graph whose vertices are the self-dual codes of length n and two codes are connected if one can be constructed from the other by the neighbor construction. We show that this graph is connected and regular with degree .
{"title":"The neighbor graph of self-dual codes over the ring of integers modulo 4","authors":"Steven T. Dougherty , Esengül Saltürk","doi":"10.1016/j.disc.2026.115018","DOIUrl":"10.1016/j.disc.2026.115018","url":null,"abstract":"<div><div>We describe the neighbor construction for self-dual codes over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and give the type of the neighbor based on the type of the code and vector <strong>v</strong> used to construct the neighbor. We define the neighbor graph of self-dual codes over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> as the graph whose vertices are the self-dual codes of length <em>n</em> and two codes are connected if one can be constructed from the other by the neighbor construction. We show that this graph is connected and regular with degree <span><math><mn>2</mn><mo>(</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></msubsup><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>4</mn><mi>k</mi><mo>)</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 7","pages":"Article 115018"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081592","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 : 2026-01-28DOI: 10.1016/j.disc.2026.115013
Chris Bispels , Matthew Cohen , Joshua Harrington , Joshua Lowrance , Kaelyn Pontes , Leif Schaumann , Tony W.H. Wong
Erdős first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.
{"title":"A further investigation on covering systems with odd moduli","authors":"Chris Bispels , Matthew Cohen , Joshua Harrington , Joshua Lowrance , Kaelyn Pontes , Leif Schaumann , Tony W.H. Wong","doi":"10.1016/j.disc.2026.115013","DOIUrl":"10.1016/j.disc.2026.115013","url":null,"abstract":"<div><div>Erdős first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 7","pages":"Article 115013"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081591","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}
A proper k-coloring of a graph is said to be odd if every non-isolated vertex has a color that appears an odd number of times on its neighborhood. Miao et al. (2024) [2] claimed that every planar graph without adjacent 3-cycles is odd 7-colorable and every triangle-free planar graph without intersecting 4-cycles is odd 5-colorable. Here, we point out that their published proof contains a fundamental flaw which affects the validity of the main results.
如果每个非孤立顶点的颜色在其邻域上出现奇数次，则称图的适当k-着色为奇数。Miao et al.(2024)[2]提出无相邻3环的平面图都是奇7色，无4环相交的无三角形平面图都是奇5色。在此，我们指出他们发表的证明存在一个影响主要结果有效性的根本性缺陷。
{"title":"A remark on a result on odd colorings of planar graphs","authors":"Dinabandhu Pradhan , Vaishali Sharma , Riste Škrekovski","doi":"10.1016/j.disc.2026.115014","DOIUrl":"10.1016/j.disc.2026.115014","url":null,"abstract":"<div><div>A proper <em>k</em>-coloring of a graph is said to be odd if every non-isolated vertex has a color that appears an odd number of times on its neighborhood. Miao et al. (2024) <span><span>[2]</span></span> claimed that every planar graph without adjacent 3-cycles is odd 7-colorable and every triangle-free planar graph without intersecting 4-cycles is odd 5-colorable. Here, we point out that their published proof contains a fundamental flaw which affects the validity of the main results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115014"},"PeriodicalIF":0.7,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079005","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 : 2026-01-27DOI: 10.1016/j.disc.2026.115012
Alice Lacaze-Masmonteil
We confirm most open cases of a conjecture that first appeared in Alspach et al. (1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of a hamiltonian decomposable directed graph G, such that is even and , with a hamiltonian decomposable directed graph H, such that , is also hamiltonian decomposable except possibly when G is a directed cycle and H is a directed graph of odd order that admits a decomposition into c directed hamiltonian cycle where c is odd and .
{"title":"Hamiltonian decompositions of the wreath product of hamiltonian decomposable digraphs","authors":"Alice Lacaze-Masmonteil","doi":"10.1016/j.disc.2026.115012","DOIUrl":"10.1016/j.disc.2026.115012","url":null,"abstract":"<div><div>We confirm most open cases of a conjecture that first appeared in Alspach et al. (1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of a hamiltonian decomposable directed graph <em>G</em>, such that <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> is even and <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>⩾</mo><mn>2</mn></math></span>, with a hamiltonian decomposable directed graph <em>H</em>, such that <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>|</mo><mo>⩾</mo><mn>4</mn></math></span>, is also hamiltonian decomposable except possibly when <em>G</em> is a directed cycle and <em>H</em> is a directed graph of odd order that admits a decomposition into <em>c</em> directed hamiltonian cycle where <em>c</em> is odd and <span><math><mn>3</mn><mo>⩽</mo><mi>c</mi><mo>⩽</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115012"},"PeriodicalIF":0.7,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078350","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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