{"title":"On the potential function σ(H,m,n) of an arbitrary bipartite graph H","authors":"Jian-Hua Yin, Kai-Xin Chang, Jia-Qi Huang","doi":"10.1016/j.dam.2024.11.027","DOIUrl":"10.1016/j.dam.2024.11.027","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>π</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>;</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> are two nonincreasing sequences of nonnegative integers. The pair <span><math><mrow><mi>π</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>;</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is said to be a <em>bigraphic pair</em> if there is a simple bipartite graph <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></mrow></mrow></math></span> with vertex bipartition <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> are the degrees of the vertices in <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span>, respectively. In this case, <span><math><mi>G</mi></math></span> is referred to as a <em>realization</em> of <span><math><mi>π</mi></math></span>. For a given bipartite graph <span><math><mrow><mi>H</mi><mo>=</mo><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></mrow></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>s</mi></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>Y</mi><mo>|</mo></mrow><mo>=</mo><mi>t</mi></mrow></math></span>, we say that <span><math><mi>π</mi></math></span> is a <em>potentially</em> <span><math><mi>H</mi></math></span><em>-bigraphic pair</em> if <span><math><mi>π</mi></math></span> has a realization <span><math><mi>G</mi></math></span> containing <span><math><mi>H</mi></math></span> as a subgraph with the <span><math><mi>s</mi></math></span> vertices of <span><math><mi>X</mi></math></span> in the part of ","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 189-194"},"PeriodicalIF":1.0,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747587","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":"Turán problem of signed graph for negative odd cycle","authors":"Junjie Wang , Yaoping Hou , Xueyi Huang","doi":"10.1016/j.dam.2024.11.024","DOIUrl":"10.1016/j.dam.2024.11.024","url":null,"abstract":"<div><div>We investigate natural Turán problems on signed graphs in this paper. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msubsup></math></span> denote the signed cycle of length <span><math><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> with one negative edge. We determine the maximum number of edges among all unbalanced signed graphs of order <span><math><mi>n</mi></math></span> with no subgraph switching to <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msubsup></math></span>, where <span><math><mrow><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>10</mn></mrow></mfrac><mo>−</mo><mn>1</mn></mrow></math></span>, and characterize the extremal signed graphs. As a by-product, we also obtain the maximum spectral radius among these signed graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 157-166"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722206","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}
Naga V.C. Gudapati, Enrico Malaguti, Michele Monaci, Paolo Paronuzzi
{"title":"A computational study on Integer Programming formulations for Hop-constrained survivable network design","authors":"Naga V.C. Gudapati, Enrico Malaguti, Michele Monaci, Paolo Paronuzzi","doi":"10.1016/j.dam.2024.11.021","DOIUrl":"10.1016/j.dam.2024.11.021","url":null,"abstract":"<div><div>We consider a Network Design problem where edges have to be activated at minimum cost while ensuring that the resulting graph contains at least <span><math><mi>k</mi></math></span> disjoint paths linking a given set of origin–destination pairs. In addition, those paths are constrained in terms of maximum number of intermediate nodes. We consider alternative Integer Programming formulations for the problem and computationally evaluate them on a large benchmark of instances having different features. Finally, we extend our analysis to the case in which the paths must be vertex disjoint.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 71-81"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722205","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}
Xiaomin Hu , Xiaowang Li , Shurong Zhang , Weihua Yang
{"title":"Subnetwork reliability analysis of star networks","authors":"Xiaomin Hu , Xiaowang Li , Shurong Zhang , Weihua Yang","doi":"10.1016/j.dam.2024.11.017","DOIUrl":"10.1016/j.dam.2024.11.017","url":null,"abstract":"<div><div>Subnetwork reliability is an important way to measure network reliability. Wu and Lati (2008) used the probability fault model to estimate an upper bound of an <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional star reliability in an <span><math><mi>n</mi></math></span>-dimensional star. In this paper, by taking into account the intersection of no more than four subnetworks, we find a lower bound of an <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional star reliability in an <span><math><mi>n</mi></math></span>-dimensional star under the probability fault model. An approximation of subnetwork reliability can be obtained by taking the mean of an upper bound and a lower bound. Based on the efficiency analysis, this result has little error and is well consistent with the exact value, especially with the expansion of network.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 180-188"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722204","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}
Michael A. Henning , Arti Pandey , Vikash Tripathi
{"title":"More on the complexity of defensive domination in graphs","authors":"Michael A. Henning , Arti Pandey , Vikash Tripathi","doi":"10.1016/j.dam.2024.11.023","DOIUrl":"10.1016/j.dam.2024.11.023","url":null,"abstract":"<div><div>In 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>, a non-empty set <span><math><mi>A</mi></math></span> of <span><math><mi>k</mi></math></span> distinct vertices, is called a <span><math><mi>k</mi></math></span>-<em>attack</em> on <span><math><mi>G</mi></math></span>. The vertices in the set <span><math><mi>A</mi></math></span> are considered to be <em>under attack</em>. A set <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> can defend or counter the attack <span><math><mi>A</mi></math></span> on <span><math><mi>G</mi></math></span> if there exists a one-to-one function <span><math><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>⟼</mo><mi>D</mi></mrow></math></span>, such that either <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>u</mi></mrow></math></span> or there is an edge between <span><math><mi>u</mi></math></span>, and its image <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, in <span><math><mi>G</mi></math></span>. A set <span><math><mi>D</mi></math></span> is called a <span><math><mi>k</mi></math></span>-<em>defensive dominating set</em> if it defends against any <span><math><mi>k</mi></math></span>-attack on <span><math><mi>G</mi></math></span>. Given 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>, the minimum <span><math><mi>k</mi></math></span>-defensive domination problem requires us to compute a minimum cardinality <span><math><mi>k</mi></math></span>-defensive dominating set of <span><math><mi>G</mi></math></span>. When <span><math><mi>k</mi></math></span> is not fixed, it is co-NP-hard to decide if <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is a <span><math><mi>k</mi></math></span>-defensive dominating set. However, when <span><math><mi>k</mi></math></span> is fixed, the decision version of the problem is NP-complete for general graphs. On the positive side, the problem can be solved in linear time when restricted to paths, cycles, co-chain, and threshold graphs for any <span><math><mi>k</mi></math></span>. This paper mainly focuses on the problem when <span><math><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></math></span> is fixed. We prove that the decision version of the problem remains NP-complete for bipartite graphs; this answers a question asked by Ekim et al. (Discrete Math. 343 (2) (2020)). We establish a lower and upper bound on the approximation ratio for the problem. Further, we show that the minimum <span><math><mi>k</mi></math></span>-defensive domination problem is APX-complete for bounded degree graphs. On the positive side, we show that the problem is efficiently solvable for complete bipartite graphs for any <span><math><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></math></span>. Towards the end, we study a relatio","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 167-179"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722207","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}
Xiaofeng Gu , Jessica Li , Eric H. Yang , William Y. Zhang
{"title":"Cyclic base ordering of certain degenerate graphs","authors":"Xiaofeng Gu , Jessica Li , Eric H. Yang , William Y. Zhang","doi":"10.1016/j.dam.2024.11.022","DOIUrl":"10.1016/j.dam.2024.11.022","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph. A cyclic base ordering of <span><math><mi>G</mi></math></span> is a cyclic ordering of elements in <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that every cyclically consecutive <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> edges form a spanning tree of <span><math><mi>G</mi></math></span>. The density of <span><math><mi>G</mi></math></span> is defined to be <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>; and <span><math><mi>G</mi></math></span> is uniformly dense if <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for every connected subgraph <span><math><mi>H</mi></math></span> of <span><math><mi>G</mi></math></span>. It was conjectured by Kajitani, Ueno and Miyano that <span><math><mi>G</mi></math></span> has a cyclic base ordering if and only if <span><math><mi>G</mi></math></span> is uniformly dense. We show that the conjecture holds for maximal 2-degenerate graphs and graphs with uniform ear decompositions. As applications, book graphs, broken fan and broken wheel graphs have cyclic base ordering. We also study cyclic base ordering of double wheel graphs and the square of cycles.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 148-156"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706126","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}
Ervin Győri , Binlong Li , Nika Salia , Casey Tompkins
{"title":"A note on universal graphs for spanning trees","authors":"Ervin Győri , Binlong Li , Nika Salia , Casey Tompkins","doi":"10.1016/j.dam.2024.11.008","DOIUrl":"10.1016/j.dam.2024.11.008","url":null,"abstract":"<div><div>Chung and Graham considered the problem of minimizing the number of edges in an <span><math><mi>n</mi></math></span>-vertex graph containing all <span><math><mi>n</mi></math></span>-vertex trees as a subgraph. They showed that such a graph has at least <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>n</mi><mo>log</mo><mi>n</mi></mrow></math></span> edges. In this note, we improve this lower estimate to <span><math><mrow><mi>n</mi><mo>log</mo><mi>n</mi><mo>−</mo><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 146-147"},"PeriodicalIF":1.0,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706125","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 new condition on dominated pair degree sum for a digraph to be supereulerian","authors":"Changchang Dong , Jixiang Meng , Juan Liu","doi":"10.1016/j.dam.2024.11.006","DOIUrl":"10.1016/j.dam.2024.11.006","url":null,"abstract":"<div><div>A digraph <span><math><mi>D</mi></math></span> is supereulerian if <span><math><mi>D</mi></math></span> contains a spanning Eulerian subdigraph. For any two vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> in a digraph <span><math><mi>D</mi></math></span>, if <span><math><mrow><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>∈</mo><mi>A</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> for some <span><math><mrow><mi>w</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span>, then we call the pair <span><math><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></mrow></math></span> dominating; if <span><math><mrow><mrow><mo>(</mo><mi>w</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><mi>A</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> for some <span><math><mrow><mi>w</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span>, then we call the pair <span><math><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></mrow></math></span> dominated. In 2015, Bang–Jensen and Maddaloni (2015) proved that if a strong digraph <span><math><mi>D</mi></math></span> with <span><math><mi>n</mi></math></span> vertices satisfies <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span> for any pair of nonadjacent vertices <span><math><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></mrow></math></span> of <span><math><mi>D</mi></math></span>, then <span><math><mi>D</mi></math></span> is supereulerian. In this paper, we study degree sum conditions only for any pair of dominated or dominating nonadjacent vertices to assure the digraph to be supereulerian, which imply the above-mentioned result.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 124-130"},"PeriodicalIF":1.0,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706128","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":"Ramsey and Gallai–Ramsey numbers for comb and sun graphs","authors":"Xiao Xu , Meiqin Wei , Hong-Jian Lai , Yaping Mao","doi":"10.1016/j.dam.2024.11.001","DOIUrl":"10.1016/j.dam.2024.11.001","url":null,"abstract":"<div><div>Given two graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the <em>Ramsey number</em> <span><math><mrow><mo>R</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined as the minimum number of vertices <span><math><mi>n</mi></math></span> such that every <span><math><mrow><mo>{</mo><mi>r</mi><mi>e</mi><mi>d</mi><mo>,</mo><mi>b</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo>}</mo></mrow></math></span>-edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> contains either a red copy of <span><math><mi>G</mi></math></span> or a blue copy of <span><math><mi>H</mi></math></span>. If <span><math><mrow><mi>G</mi><mo>≅</mo><mi>H</mi></mrow></math></span>, then we write <span><math><mrow><mo>R</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for short. For any positive integer <span><math><mi>k</mi></math></span>, the Gallai–Ramsey number <span><math><mrow><msub><mrow><mo>gr</mo></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>:</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the minimum number of vertices <span><math><mi>n</mi></math></span> such that any exact <span><math><mi>k</mi></math></span>-edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> contains either a rainbow copy of <span><math><mi>G</mi></math></span> or a monochromatic copy of <span><math><mi>H</mi></math></span>. In this paper, we give exact values or upper and lower bounds for Ramsey numbers <span><math><mrow><mo>R</mo><mrow><mo>(</mo><mi>C</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mo>R</mo><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mo>R</mo><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><mi>C</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and Gallai–Ramsey numbers <span><math><mrow><msub><mrow><mo>gr</mo></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>:</mo><mi>C</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mo>gr</mo></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>C</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are the comb and sun graphs, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 131-145"},"PeriodicalIF":1.0,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706559","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":"Bounds for eccentricity-based parameters of graphs","authors":"Yunfang Tang , Xuli Qi , Douglas B. West","doi":"10.1016/j.dam.2024.11.004","DOIUrl":"10.1016/j.dam.2024.11.004","url":null,"abstract":"<div><div>The <em>eccentricity</em> of a vertex <span><math><mi>u</mi></math></span> in a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, is the maximum distance from <span><math><mi>u</mi></math></span> to other vertices in <span><math><mi>G</mi></math></span>. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity indices, denoted by <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, respectively. These are defined by <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><msubsup><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. We study lower and upper bounds on these parameters among <span><math><mi>n</mi></math></span>-vertex connected graphs with fixed diameter, chromatic number, clique number, or matching number. Most of the bounds are sharp, with the corresponding extremal graphs characterized.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 109-123"},"PeriodicalIF":1.0,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706124","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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