Pub Date : 2026-03-31Epub Date: 2025-12-10DOI: 10.1016/j.dam.2025.12.002
Yanhong Chen , Anshui Li , Biao Wu , Huajun Zhang
Two families and are called cross--intersecting if for all , . Let , and be positive integers such that and . In this paper, we will determine the upper bound of for cross-2-intersecting families and . The structures of the extremal families attaining the upper bound are also characterized. The similar result obtained by Tokushige can be considered as a special case of ours when , but under a more strong condition . Moreover, combined with the results obtained in this paper, the complicated extremal structures attaining the upper bound for nontrivial cases can be relatively easy to reach with similar techniques.
{"title":"On cross-2-intersecting families","authors":"Yanhong Chen , Anshui Li , Biao Wu , Huajun Zhang","doi":"10.1016/j.dam.2025.12.002","DOIUrl":"10.1016/j.dam.2025.12.002","url":null,"abstract":"<div><div>Two families <span><math><mrow><mi>A</mi><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> and <span><math><mrow><mi>B</mi><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><mi>ℓ</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> are called cross-<span><math><mi>t</mi></math></span>-intersecting if <span><math><mrow><mrow><mo>|</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>|</mo></mrow><mo>≥</mo><mi>t</mi></mrow></math></span> for all <span><math><mrow><mi>A</mi><mo>∈</mo><mi>A</mi></mrow></math></span>, <span><math><mrow><mi>B</mi><mo>∈</mo><mi>B</mi></mrow></math></span>. Let <span><math><mi>n</mi></math></span>, <span><math><mi>k</mi></math></span> and <span><math><mi>ℓ</mi></math></span> be positive integers such that <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn><mo>.</mo><mn>38</mn><mi>ℓ</mi></mrow></math></span> and <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. In this paper, we will determine the upper bound of <span><math><mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mrow><mo>|</mo><mi>B</mi><mo>|</mo></mrow></mrow></math></span> for cross-2-intersecting families <span><math><mrow><mi>A</mi><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> and <span><math><mrow><mi>B</mi><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><mi>ℓ</mi></mrow></mfrac></mrow></mfenced></mrow></math></span>. The structures of the extremal families attaining the upper bound are also characterized. The similar result obtained by Tokushige can be considered as a special case of ours when <span><math><mrow><mi>k</mi><mo>=</mo><mi>ℓ</mi></mrow></math></span>, but under a more strong condition <span><math><mrow><mi>n</mi><mo>></mo><mn>3</mn><mo>.</mo><mn>42</mn><mi>k</mi></mrow></math></span>. Moreover, combined with the results obtained in this paper, the complicated extremal structures attaining the upper bound for nontrivial cases can be relatively easy to reach with similar techniques.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 259-271"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737339","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 : 2026-03-31Epub Date: 2026-01-12DOI: 10.1016/j.dam.2025.12.071
Zihan Ning , Ran Ma , Zhengwen He , Xiaoyan Zhang
In project implementation, uncertainty factors have significant impacts. How to effectively safeguard staff working conditions while ensuring timely project delivery under uncertain environments has emerged as a critical issue urgently requiring resolution in the project management domain. The Multi-Skill Resource-Constrained Project Scheduling Problem (MS-RCPSP), as a typical combinatorial optimization challenge, provides theoretical foundations for addressing the issues. This problem can be abstracted as a complex multi-layer network graph optimization model. This study focuses on robust project scheduling under multi-skill resource constraints, aiming to generate proactive optimization schedules that simultaneously minimize staff maximum fatigue accumulation and maximize robustness. To accurately reflect realistic execution environments, this research incorporates activity uncertainty into staff fatigue levels considerations within the model construction, thereby better characterizing the actual impact of uncertain factors on staff fatigue levels. In terms of algorithmic design, this paper introduces priority selection strategies during the schedule generation process to enhance solution quality, and develops an Improved Non-dominated Sorting Genetic Algorithm II (INSGA-II) featuring dual-mode adaptive selection mechanisms and diversity-driven elite retention adjustments. Experimental results demonstrate that through multi-metric comparisons, the proposed comprehensive algorithm significantly outperforms traditional methods. Furthermore, this study conducts in-depth experimental analysis and interpretation regarding solution robustness and the improved fatigue function, validating the effectiveness and practicality of the proposed methodology.
{"title":"Robust optimization of multi-skill resource-constrained project networks considering staff fatigue under uncertainty","authors":"Zihan Ning , Ran Ma , Zhengwen He , Xiaoyan Zhang","doi":"10.1016/j.dam.2025.12.071","DOIUrl":"10.1016/j.dam.2025.12.071","url":null,"abstract":"<div><div>In project implementation, uncertainty factors have significant impacts. How to effectively safeguard staff working conditions while ensuring timely project delivery under uncertain environments has emerged as a critical issue urgently requiring resolution in the project management domain. The Multi-Skill Resource-Constrained Project Scheduling Problem (MS-RCPSP), as a typical combinatorial optimization challenge, provides theoretical foundations for addressing the issues. This problem can be abstracted as a complex multi-layer network graph optimization model. This study focuses on robust project scheduling under multi-skill resource constraints, aiming to generate proactive optimization schedules that simultaneously minimize staff maximum fatigue accumulation and maximize robustness. To accurately reflect realistic execution environments, this research incorporates activity uncertainty into staff fatigue levels considerations within the model construction, thereby better characterizing the actual impact of uncertain factors on staff fatigue levels. In terms of algorithmic design, this paper introduces priority selection strategies during the schedule generation process to enhance solution quality, and develops an Improved Non-dominated Sorting Genetic Algorithm II (INSGA-II) featuring dual-mode adaptive selection mechanisms and diversity-driven elite retention adjustments. Experimental results demonstrate that through multi-metric comparisons, the proposed comprehensive algorithm significantly outperforms traditional methods. Furthermore, this study conducts in-depth experimental analysis and interpretation regarding solution robustness and the improved fatigue function, validating the effectiveness and practicality of the proposed methodology.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 383-399"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976748","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 : 2026-03-31Epub Date: 2025-12-15DOI: 10.1016/j.dam.2025.12.011
Olivier Baudon , Julien Bensmail , Morgan Boivin , Igor Grzelec , Clara Marcille
The Strong -Conjecture asks whether, for all connected graphs different from and , we can assign to edges red and blue labels with value 1 or 2 so that no two adjacent vertices have the same sum of incident red labels or the same sum of incident blue labels. This conjecture, which can be perceived as a generalisation of the so-called 1–2–3 Conjecture, as, thus far, been proved only for a handful number of graph classes. In this work, we prove the Strong -Conjecture holds for more classes of graphs. In particular, we prove the conjecture for cacti, subcubic outerplanar graphs, graphs with maximum average degree less than , and some Halin graphs, among others.
{"title":"The strong (2,2)-Conjecture for more classes of graphs","authors":"Olivier Baudon , Julien Bensmail , Morgan Boivin , Igor Grzelec , Clara Marcille","doi":"10.1016/j.dam.2025.12.011","DOIUrl":"10.1016/j.dam.2025.12.011","url":null,"abstract":"<div><div>The Strong <span><math><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-Conjecture asks whether, for all connected graphs different from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, we can assign to edges red and blue labels with value 1 or 2 so that no two adjacent vertices have the same sum of incident red labels or the same sum of incident blue labels. This conjecture, which can be perceived as a generalisation of the so-called 1–2–3 Conjecture, as, thus far, been proved only for a handful number of graph classes. In this work, we prove the Strong <span><math><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-Conjecture holds for more classes of graphs. In particular, we prove the conjecture for cacti, subcubic outerplanar graphs, graphs with maximum average degree less than <span><math><mfrac><mrow><mn>9</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>, and some Halin graphs, among others.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 337-354"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790312","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 : 2026-03-31Epub Date: 2025-12-12DOI: 10.1016/j.dam.2025.12.014
Chunli Deng , Junchen Dong , Haifeng Li
For -uniform hypergraphs, this paper establishes several new bounds on the spectral radii of the adjacency and signless Laplacian tensors. These bounds are expressed by vertex degrees and average 2-degrees of hypergraphs. Furthermore, the bounds are compared with the known results, and the example shows that our bounds are better in some cases. As applications, the spectral radii of the generalized power hypergraphs of stars and cycles are presented, respectively.
{"title":"Several bounds on the spectral radius of uniform hypergraphs","authors":"Chunli Deng , Junchen Dong , Haifeng Li","doi":"10.1016/j.dam.2025.12.014","DOIUrl":"10.1016/j.dam.2025.12.014","url":null,"abstract":"<div><div>For <span><math><mi>k</mi></math></span>-uniform hypergraphs, this paper establishes several new bounds on the spectral radii of the adjacency and signless Laplacian tensors. These bounds are expressed by vertex degrees and average 2-degrees of hypergraphs. Furthermore, the bounds are compared with the known results, and the example shows that our bounds are better in some cases. As applications, the spectral radii of the generalized power hypergraphs of stars and cycles are presented, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 317-326"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737340","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 : 2026-03-31Epub 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":"2026-03-31","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}
Pub Date : 2026-03-31Epub 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":"2026-03-31","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 : 2026-03-31Epub Date: 2026-01-19DOI: 10.1016/j.dam.2026.01.012
Ruiqing Feng, Qi Yan, Xuan Zheng
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose a fundamental characterization problem: identify all connected circle graphs whose partial Petrial polynomial is binomial. In this paper, we solve this open problem in terms of local complementation and prove that for connected circle graphs, the binomial property holds precisely when the graph is a path.
{"title":"Characterizing circle graphs with binomial partial Petrial polynomials","authors":"Ruiqing Feng, Qi Yan, Xuan Zheng","doi":"10.1016/j.dam.2026.01.012","DOIUrl":"10.1016/j.dam.2026.01.012","url":null,"abstract":"<div><div>The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose a fundamental characterization problem: identify all connected circle graphs whose partial Petrial polynomial is binomial. In this paper, we solve this open problem in terms of local complementation and prove that for connected circle graphs, the binomial property holds precisely when the graph is a path.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 411-416"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023054","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 : 2026-03-31Epub Date: 2026-01-09DOI: 10.1016/j.dam.2025.12.057
Xiaotong Gu, Hongzhi Deng, Yuting Tian, Jianhua Tu
In this paper, we determine the maximum number of maximal independent sets for three families of graphs. The first family comprises all 2-connected graphs with matching number . The second family consists of all unicyclic graphs with matching number . The third family encompasses all graphs on vertices with matching number where ; note that the case has been settled in previous work.
{"title":"Maximizing the number of maximal independent sets in graphs with a given matching number","authors":"Xiaotong Gu, Hongzhi Deng, Yuting Tian, Jianhua Tu","doi":"10.1016/j.dam.2025.12.057","DOIUrl":"10.1016/j.dam.2025.12.057","url":null,"abstract":"<div><div>In this paper, we determine the maximum number of maximal independent sets for three families of graphs. The first family comprises all 2-connected graphs with matching number <span><math><mi>t</mi></math></span>. The second family consists of all unicyclic graphs with matching number <span><math><mi>t</mi></math></span>. The third family encompasses all graphs on <span><math><mi>n</mi></math></span> vertices with matching number <span><math><mi>t</mi></math></span> where <span><math><mrow><mn>2</mn><mi>t</mi><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>3</mn><mi>t</mi></mrow></math></span>; note that the case <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn><mi>t</mi></mrow></math></span> has been settled in previous work.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 368-382"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924781","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 : 2026-03-31Epub Date: 2026-01-15DOI: 10.1016/j.dam.2026.01.002
Yushuang Mou , Qiang Sun , Chao Zhang
Weighted signed networks capture both positive and negative relationships between individuals, with link weights representing the intensity of these relationships. We model cooperation in such networks as a cooperative game restricted by a weighted signed network. To address the distribution problem in these games, we introduce the weighted signed Myerson value (WS-Myerson value), which is grounded in structural balance theory and incorporates the minimum cost required to achieve balance within the network. We prove that the WS-Myerson value is uniquely determined by the axioms of component efficiency, fairness for conflict players, and marginality.
{"title":"The Myerson value for games with weighted signed networks","authors":"Yushuang Mou , Qiang Sun , Chao Zhang","doi":"10.1016/j.dam.2026.01.002","DOIUrl":"10.1016/j.dam.2026.01.002","url":null,"abstract":"<div><div>Weighted signed networks capture both positive and negative relationships between individuals, with link weights representing the intensity of these relationships. We model cooperation in such networks as a cooperative game restricted by a weighted signed network. To address the distribution problem in these games, we introduce the weighted signed Myerson value (WS-Myerson value), which is grounded in structural balance theory and incorporates the minimum cost required to achieve balance within the network. We prove that the WS-Myerson value is uniquely determined by the axioms of component efficiency, fairness for conflict players, and marginality.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 400-410"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976747","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 : 2026-03-31Epub Date: 2025-12-04DOI: 10.1016/j.dam.2025.10.062
David Avis
A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron in can be described by an consisting of half spaces or equivalently by a -representation consisting of the convex hull of a set of vertices and extreme rays. We can define matrices and , each with columns, that encode these representations. Define polyhedron by setting . We show that is the polar of . Call -symmetric if is pointed and in turn encodes . It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are -symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is -symmetric if and only if it contains the origin. We prove this using Minkowski’s bipolar equation and discuss implications for the vertex and facet enumeration problems.
{"title":"HV-symmetric polyhedra and bipolarity","authors":"David Avis","doi":"10.1016/j.dam.2025.10.062","DOIUrl":"10.1016/j.dam.2025.10.062","url":null,"abstract":"<div><div>A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron <span><math><mi>P</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> can be described by an <span><math><mrow><mi>H</mi><mo>−</mo><mi>r</mi><mi>e</mi><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></math></span>\u0000 <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> consisting of half spaces or equivalently by a <span><math><mi>V</mi></math></span>-representation <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> consisting of the convex hull of a set of vertices and extreme rays. We can define matrices <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span>, each with <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> columns, that encode these representations. Define polyhedron <span><math><mi>Q</mi></math></span> by setting <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span>. We show that <span><math><mi>Q</mi></math></span> is the polar of <span><math><mi>P</mi></math></span>. Call <span><math><mi>P</mi></math></span>\u0000 <span><math><mrow><mi>H</mi><mi>V</mi></mrow></math></span>-symmetric if <span><math><mi>Q</mi></math></span> is pointed and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> in turn encodes <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span>. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are <span><math><mrow><mi>H</mi><mi>V</mi></mrow></math></span>-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is <span><math><mrow><mi>H</mi><mi>V</mi></mrow></math></span>-symmetric if and only if it contains the origin. We prove this using Minkowski’s bipolar equation and discuss implications for the vertex and facet enumeration problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 210-213"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685322","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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