Discrete MathematicsPub Date : 2026-07-01Epub Date: 2026-01-29DOI: 10.1016/j.disc.2026.115019
Zixuan Yang , Hongliang Lu , Shenggui Zhang
{"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":"2026-07-01","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}
Discrete MathematicsPub Date : 2026-07-01Epub Date: 2026-01-28DOI: 10.1016/j.disc.2026.115017
L. Sunil Chandran , Jinia Ghosh
{"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":"2026-07-01","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}
Discrete MathematicsPub Date : 2026-07-01Epub Date: 2026-01-28DOI: 10.1016/j.disc.2026.115018
Steven T. Dougherty , Esengül Saltürk
{"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":"2026-07-01","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}
Discrete MathematicsPub Date : 2026-07-01Epub 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
{"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":"2026-07-01","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}
Discrete MathematicsPub Date : 2026-07-01Epub Date: 2026-01-27DOI: 10.1016/j.disc.2026.115016
Lele Liu , Bo Ning
{"title":"Spectral Turán-type problems on sparse spanning graphs","authors":"Lele Liu , Bo Ning","doi":"10.1016/j.disc.2026.115016","DOIUrl":"10.1016/j.disc.2026.115016","url":null,"abstract":"<div><div>Let <em>F</em> be a graph, and let <span><math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, and <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> denote the classes of graphs that attain, respectively, the maximum number of edges, the maximum adjacency spectral radius, and the maximum signless Laplacian spectral radius over all <em>n</em>-vertex graphs that do not contain <em>F</em> as a subgraph. A fundamental problem in spectral extremal graph theory is to characterize all graphs <em>F</em> for which <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> when <em>n</em> is sufficiently large. Establishing the conjecture of Cioabă et al. (2022) <span><span>[10]</span></span>, Wang et al. (2023) <span><span>[54]</span></span> proved that: for any graph <em>F</em> such that the graphs in <span><math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> are Turán graphs plus <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> edges, <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> for sufficiently large <em>n</em>. In addition, another interesting problem in spectral extremal graph theory is to characterize all graphs <em>F</em> such that <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> for sufficiently large <em>n</em>.</div><div>In this paper, we give new contribution to the problems mentioned above. First, we present a substantial collection of examples of graphs <em>F</em> for which <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> holds when <em>n</em> is sufficiently large, focusing on <em>n</em>-vertex graph <em>F</em> with no isolated vertices and maximum degree <span><math><mi>Δ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>/</mo><mn>40</mn></math></span>. Second, under the same conditions on <em>F</em>, we prove that <span><math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 7","pages":"Article 115016"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049174","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}
Discrete MathematicsPub Date : 2026-06-01Epub Date: 2026-01-29DOI: 10.1016/j.disc.2026.115024
Jing Yu , Jie-Xiang Zhu
{"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":"2026-06-01","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}
Discrete MathematicsPub Date : 2026-06-01Epub Date: 2026-01-19DOI: 10.1016/j.disc.2026.115006
Yusuke Ide , Takashi Komatsu , Norio Konno , Iwao Sato
{"title":"A Metzler matrix of a group covering of a digraph","authors":"Yusuke Ide , Takashi Komatsu , Norio Konno , Iwao Sato","doi":"10.1016/j.disc.2026.115006","DOIUrl":"10.1016/j.disc.2026.115006","url":null,"abstract":"<div><div>We present a decomposition formula for the determinant of a Metzler matrix <span><math><mi>A</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of a group covering <em>H</em> of a digraph <em>D</em>. Furthermore, we introduce an <em>L</em>-function of <em>D</em> with respect to its Metzler matrix <span><math><mi>A</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, and present a determinant expression of it. As a corollary, we present a decomposition formula for the determinant of a Metzler matrix <span><math><mi>A</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of a group covering <em>H</em> of <em>D</em> by its <em>L</em>-functions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115006"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038451","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}
Discrete MathematicsPub Date : 2026-06-01Epub Date: 2026-01-16DOI: 10.1016/j.disc.2026.114996
Tianjiao Dai , Jie Hu , Hao Li , Shun-ichi Maezawa
{"title":"On DP-coloring of outerplanar graphs","authors":"Tianjiao Dai , Jie Hu , Hao Li , Shun-ichi Maezawa","doi":"10.1016/j.disc.2026.114996","DOIUrl":"10.1016/j.disc.2026.114996","url":null,"abstract":"<div><div>The notion of DP-coloring was introduced by Dvořák and Postle which is a generalization of list coloring. A DP-coloring of a graph <em>G</em> reduces the problem of finding a proper coloring of <em>G</em> from a given list <em>L</em> to the problem of finding a “large” independent set in an auxiliary graph <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span>-cover with a vertex set <span><math><mo>{</mo><mo>(</mo><mi>v</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>:</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>c</mi><mo>∈</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo></math></span>. Hutchinson (Journal of Graph Theory, 2008) showed that<ul><li><span>•</span><span><div>if a 2-connected bipartite outerplanar graph <em>G</em> has a list of colors <span><math><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> for each vertex <em>v</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mi>min</mi><mo></mo><mo>{</mo><msub><mrow><mi>deg</mi></mrow><mrow><mi>G</mi></mrow></msub><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mn>4</mn><mo>}</mo></math></span>, then <em>G</em> is <em>L</em>-colorable; and</div></span></li><li><span>•</span><span><div>if a 2-connected maximal outerplanar graph <em>G</em> with at least four vertices has a list of colors <span><math><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> for each vertex <em>v</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mi>min</mi><mo></mo><mo>{</mo><msub><mrow><mi>deg</mi></mrow><mrow><mi>G</mi></mrow></msub><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mn>5</mn><mo>}</mo></math></span>, then <em>G</em> is <em>L</em>-colorable.</div></span></li></ul> In this paper, we study whether bounds of Hutchinson's results hold for DP-coloring. We obtain that the first one is not sufficient for DP-coloring while the second one is sufficient.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114996"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980013","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}
Discrete MathematicsPub Date : 2026-06-01Epub Date: 2026-01-13DOI: 10.1016/j.disc.2026.114997
Kiyoshi Ando , Yoshimi Egawa
{"title":"A constructive characterization of 4-connected 4-regular graphs","authors":"Kiyoshi Ando , Yoshimi Egawa","doi":"10.1016/j.disc.2026.114997","DOIUrl":"10.1016/j.disc.2026.114997","url":null,"abstract":"<div><div>In this paper, we give a constructive characterization of 4-connected 4-regular graphs. Two edges of a graph are said to be “independent” if they have no common end vertex. Let <em>G</em> be a 4-connected 4-regular graph. We consider the following three operations on <em>G</em>: choose two independent edges of <em>G</em>, subdivide once, and identify the two new vertices (we call this operation “edge-binding”); delete a vertex <em>x</em> from <em>G</em>, add <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> to <span><math><mi>G</mi><mo>−</mo><mi>x</mi></math></span>, and add a perfect matching between <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> (we call this operation “<span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-expanding”); delete two independent edges <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> from <em>G</em>, add <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> to <span><math><mi>G</mi><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and add a perfect matching between <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>∪</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> (we call this operation “<span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-edge-binding”). In this paper, we prove that every 4-connected 4-regular graph can be obtained from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span> by repeated applications of edge-bindings, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-expandings and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-edge-bindings.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114997"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980074","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}
Discrete MathematicsPub Date : 2026-06-01Epub Date: 2026-01-08DOI: 10.1016/j.disc.2026.114979
Ran Chen, Baogang Xu
{"title":"Structure and linear-Pollyanna for some square-free graphs","authors":"Ran Chen, Baogang Xu","doi":"10.1016/j.disc.2026.114979","DOIUrl":"10.1016/j.disc.2026.114979","url":null,"abstract":"<div><div>We use <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> to denote a path and a cycle on <em>t</em> vertices, respectively. A <em>bull</em> is a graph consisting of a triangle with two disjoint pendant edges, a <em>hammer</em> is a graph obtained by identifying an endvertex of a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> with a vertex of a triangle. A class <span><math><mi>F</mi></math></span> is <em>χ</em>-bounded if there is a function <em>f</em> such that <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> for all induced subgraphs <em>G</em> of a graph in <span><math><mi>F</mi></math></span>. A class <span><math><mi>C</mi></math></span> of graphs is <em>Pollyanna</em> (resp. <em>linear-Pollyanna</em>) if <span><math><mi>C</mi><mo>∩</mo><mi>F</mi></math></span> is polynomially (resp. linearly) <em>χ</em>-bounded for every <em>χ</em>-bounded class <span><math><mi>F</mi></math></span> of graphs. Chudnovsky et al. <span><span>[6]</span></span> showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let <em>G</em> be a connected graph with no clique cutsets and no universal vertices. In this paper, we show that <em>G</em> is <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, hammer)-free if and only if it has girth at least 5, and <em>G</em> is <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, bull)-free graphs and <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, hammer)-free graphs are linear-Pollyannas. We also show that the class of (bull, diamond)-free graphs is linear-Pollyanna.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114979"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928976","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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