{"title":"Shifted and threshold matroids","authors":"Ethan Partida","doi":"10.1016/j.dam.2025.08.038","DOIUrl":"10.1016/j.dam.2025.08.038","url":null,"abstract":"<div><div>We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of our characterization of threshold matroids: We give a formula for the number of isomorphism classes of threshold matroids on a ground set of size <span><math><mi>n</mi></math></span>. This enumeration shows that almost all shifted matroids are not threshold. We also present a polynomial-time algorithm to check if a matroid is threshold and provide alternative and simplified proofs of some of the main results of Deza and Onn.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 522-537"},"PeriodicalIF":1.0,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896187","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}
Shantanu Das , Nikos Giachoudis , Flaminia L. Luccio , Euripides Markou
{"title":"On the Broadcast problem for mobile agents in dynamic networks","authors":"Shantanu Das , Nikos Giachoudis , Flaminia L. Luccio , Euripides Markou","doi":"10.1016/j.dam.2025.08.031","DOIUrl":"10.1016/j.dam.2025.08.031","url":null,"abstract":"<div><div>We study the standard communication problem of broadcast for mobile agents moving in a network, where a single agent called source, has to transmit a vital information to all other agents in the network. The agents move autonomously in the network and can communicate with other agents only when they meet at a node. Previous studies of this problem were restricted to static networks while, in this paper, we consider the problem in dynamic networks modeled as an evolving graph. The dynamicity of the graph is unknown to the agents; in each round an adversary selects which edges of the graph are available, and an agent can choose to traverse one of the available edges adjacent to its current location. The only restriction on the adversary is that the subgraph of available edges in each round must span all nodes; in other words the evolving graph is constantly connected. The agents have global visibility allowing them to see the location of all agents in the graph and move accordingly. Depending on the topology of the underlying graph, we determine the minimum value of <span><math><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></math></span>, such that the broadcast from a source agent to <span><math><mi>k</mi></math></span> other agents can be solved in dynamic networks. While <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span> agents are sufficient for ring networks, much larger teams of agents are necessary for denser graphs such as grid graphs and hypercubes, and finally for complete graphs of <span><math><mi>n</mi></math></span> nodes <span><math><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span> agents are necessary and sufficient. We show lower bounds on the number of agents and provide algorithms for solving broadcast using the minimum number of agents, for various topologies. These results show how the connectivity of the underlying graph affects the communication capability of a team of mobile agents in constantly connected dynamic networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 194-208"},"PeriodicalIF":1.0,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896140","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}
Irene Heinrich , Ferdinand Ihringer , Simon Raßmann , Lena Volk
{"title":"On the twin-width of near-regular graphs","authors":"Irene Heinrich , Ferdinand Ihringer , Simon Raßmann , Lena Volk","doi":"10.1016/j.dam.2025.07.044","DOIUrl":"10.1016/j.dam.2025.07.044","url":null,"abstract":"<div><div>Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory—particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known.</div><div>This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and <span><math><mi>d</mi></math></span>-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 177-193"},"PeriodicalIF":1.0,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144892878","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}
{"title":"A decomposition structure of resonance graphs that are daisy cubes","authors":"Zhongyuan Che , Zhibo Chen","doi":"10.1016/j.dam.2025.08.030","DOIUrl":"10.1016/j.dam.2025.08.030","url":null,"abstract":"<div><div>It has recently been shown in Brezovnik et al. (2025) that the resonance graph of a plane elementary bipartite graph <span><math><mi>G</mi></math></span> with more than two vertices is a daisy cube if and only if <span><math><mi>G</mi></math></span> is peripherally 2-colorable. Let <span><math><mi>G</mi></math></span> be a peripherally 2-colorable graph and <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be its resonance graph. We provide a decomposition structure of <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with respect to an arbitrary finite face of <span><math><mi>G</mi></math></span> together with a proper labeling for the vertex set of <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. An algorithm is obtained to generate a binary coding for all perfect matchings of <span><math><mi>G</mi></math></span> which induces an isometric embedding of <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> as a daisy cube into an <span><math><mi>n</mi></math></span>-dimensional hypercube, where <span><math><mi>n</mi></math></span> is the isometric dimension of <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We conclude the paper with an easy conversion between two binary codings for all perfect matchings of <span><math><mi>G</mi></math></span> which induce distinct structures on <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>: one as a daisy cube and the other as a finite distributive lattice, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 529-540"},"PeriodicalIF":1.0,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144890316","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}
{"title":"Some sufficient conditions for a graph to be strong star factor","authors":"Fengyun Ren , Shumin Zhang","doi":"10.1016/j.dam.2025.07.034","DOIUrl":"10.1016/j.dam.2025.07.034","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and let <span><math><mrow><mi>r</mi><mo>⩾</mo><mn>2</mn></mrow></math></span> be an integer. We call a spanning subgraph <span><math><mi>H</mi></math></span> of a graph <span><math><mi>G</mi></math></span> a strong <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>-factor if every component of <span><math><mi>H</mi></math></span> is isomorphic to an element of <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span> and is an induced subgraph of <span><math><mi>G</mi></math></span>. A <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>-factor of <span><math><mi>G</mi></math></span> is a spanning subgraph of <span><math><mi>G</mi></math></span>, in which each component is isomorphic to a member in <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>. If <span><math><mi>G</mi></math></span> contains a strong <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>-factor, then <span><math><mi>G</mi></math></span> contains a <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>-factor. In this paper, through the typical spectral techniques, we establish respectively a lower bound on the size and a lower bound on the spectral radius to ensure whether or not a graph admits a strong <span><math><mrow><mo>{</mo><msub><mrow><mi>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 508-521"},"PeriodicalIF":1.0,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886780","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}
{"title":"Extremal trees with fixed order and diameter for average size of maximal matchings","authors":"Lukai Sui, Qiuli Li","doi":"10.1016/j.dam.2025.08.016","DOIUrl":"10.1016/j.dam.2025.08.016","url":null,"abstract":"<div><div>This paper characterizes the classes of graphs whose average sizes of maximal matchings achieve the extremum in trees with fixed order and diameter. Regarding the conjecture posed by Engbers and Erey about the minimum case, we confirm it for trees of fixed order with diameters 6, 7 and 8. Furthermore, we also confirm the conjecture for the maximum case in trees of fixed order with diameter 6. Our proofs base on a meticulous analysis of graph structures and a thorough exploration of matching theory. By studying trees with different diameters, we discover some interesting properties and patterns, which provide insights for further research.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 518-528"},"PeriodicalIF":1.0,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865190","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}
Anna Coleman , Gabrielle Fischberg , Charles Gong , Joshua Harrington , Tony W.H. Wong
{"title":"Paired (n−1)-to-(n−1) disjoint path covers in bipartite transposition-like graphs","authors":"Anna Coleman , Gabrielle Fischberg , Charles Gong , Joshua Harrington , Tony W.H. Wong","doi":"10.1016/j.dam.2025.07.038","DOIUrl":"10.1016/j.dam.2025.07.038","url":null,"abstract":"<div><div>A paired <span><math><mi>k</mi></math></span>-to-<span><math><mi>k</mi></math></span> disjoint path cover of a graph <span><math><mi>G</mi></math></span> is a collection of pairwise disjoint path subgraphs <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> such that each <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has prescribed vertices <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> as endpoints and the union of <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> contains all vertices of <span><math><mi>G</mi></math></span>. In this paper, we introduce bipartite transposition-like graphs, which are inductively constructed from lower ranked bipartite transposition-like graphs. We show that every rank <span><math><mi>n</mi></math></span> bipartite transposition-like graph <span><math><mi>G</mi></math></span> admit a paired <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-to-<span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> disjoint path cover for all choices of <span><math><mrow><mi>S</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><mi>T</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow></math></span>, provided that <span><math><mi>S</mi></math></span> is in one partite set of <span><math><mi>G</mi></math></span> and <span><math><mi>T</mi></math></span> is in the other.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 449-461"},"PeriodicalIF":1.0,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879933","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}
{"title":"On semi-transitive orientability of circulant graphs","authors":"Eshwar Srinivasan, Ramesh Hariharasubramanian","doi":"10.1016/j.dam.2025.08.025","DOIUrl":"10.1016/j.dam.2025.08.025","url":null,"abstract":"<div><div>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> is said to be <em>word-representable</em> if a word <span><math><mi>w</mi></math></span> can be formed using the letters of the alphabet <span><math><mi>V</mi></math></span> such that for every pair of vertices <span><math><mi>x</mi></math></span> and <span><math><mi>y</mi></math></span>, <span><math><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>E</mi></mrow></math></span> if and only if <span><math><mi>x</mi></math></span> and <span><math><mi>y</mi></math></span> alternate in <span><math><mi>w</mi></math></span>. A <em>semi-transitive</em> orientation is an acyclic directed graph where for any directed path <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>→</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mo>⋯</mo><mo>→</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> either there is no arc between <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> or for all <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></math></span> there is an arc between <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. An undirected graph is semi-transitive if it admits a semi-transitive orientation. For given positive integers <span><math><mrow><mi>n</mi><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, we consider the undirected circulant graph with set of vertices <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></math></span> and the set of edges <span><math><mrow><mo>{</mo><mi>i</mi><mi>j</mi><mo>∣</mo><mrow><mo>|</mo><mi>i</mi><mo>−</mo><mi>j</mi><mo>|</mo></mrow><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is in <span><math><mrow><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>}</mo></mrow></math></span>, where <span><math><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo>&","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 498-509"},"PeriodicalIF":1.0,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865188","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}
{"title":"On Hamiltonian bypasses in orgraphs with large semi-degrees","authors":"Samvel Kh. Darbinyan","doi":"10.1016/j.dam.2025.08.010","DOIUrl":"10.1016/j.dam.2025.08.010","url":null,"abstract":"<div><div>A Hamiltonian path in a digraph <span><math><mi>D</mi></math></span> in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. A digraph with at most one arc between any pair of vertices is called an oriented graph (or orgraph, for short). Let <span><math><mi>D</mi></math></span> be an orgraph of order <span><math><mi>p</mi></math></span> with minimum semi-degree at least <span><math><mrow><mrow><mo>⌊</mo><mi>p</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span>. In this paper, we show that if <span><math><mrow><mi>p</mi><mo>≥</mo><mn>10</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>≠</mo><mn>11</mn></mrow></math></span>, then <span><math><mi>D</mi></math></span> contains a Hamiltonian bypass. We present examples of orgraphs which shows that this result is sharp in the following sense: both lower bounds <span><math><mrow><mrow><mo>⌊</mo><mi>p</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> and 10 are tight.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 510-517"},"PeriodicalIF":1.0,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865189","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}
{"title":"Firefighting with a distance-based restriction","authors":"Andrea C. Burgess , John Marcoux , David A. Pike","doi":"10.1016/j.dam.2025.08.026","DOIUrl":"10.1016/j.dam.2025.08.026","url":null,"abstract":"<div><div>In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph <span><math><mi>G</mi></math></span> and then <span><math><mi>k</mi></math></span> firefighters protect <span><math><mi>k</mi></math></span> vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect <span><math><mi>k</mi></math></span> vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. A common objective with respect to some infinite graph <span><math><mi>G</mi></math></span> is to determine how many firefighters are necessary to stop the fire from spreading after a finite number of turns, commonly referred to as <em>containing</em> the fire. We introduce the concept of <em>distance-restricted firefighting</em> where the firefighters’ movement is restricted so they can only move up to some fixed distance <span><math><mi>d</mi></math></span> per turn rather than being able to move without restriction. We establish some general properties of this new game in contrast to properties of the original game, and we investigate specific cases of the distance-restricted game on the infinite square, strong, and hexagonal grids. We conjecture that two firefighters are insufficient on the infinite square grid when <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span>, and we pose some questions about how many firefighters are required in general when <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 480-491"},"PeriodicalIF":1.0,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867378","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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