Discrete Applied Mathematics最新文献
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Scheduling sports tournaments with two court types
安排两种场地类型的体育比赛
IF 1
3区 数学
Sigrid Knust , Melissa Koch , Xuan Thanh Le
{"title":"Scheduling sports tournaments with two court types","authors":"Sigrid Knust , Melissa Koch , Xuan Thanh Le","doi":"10.1016/j.dam.2025.07.018","DOIUrl":"10.1016/j.dam.2025.07.018","url":null,"abstract":"<div><div>In this paper, we introduce a new variant of a sports tournament scheduling problem where additionally courts of two different types are considered. The goal is to find a schedule for a single round robin tournament where the total number of consecutive matches on the same court type for the players is minimized. We propose efficient construction methods to obtain optimal solutions for different cases.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 404-426"},"PeriodicalIF":1.0,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724956","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}
引用次数: 0
The edge metric dimensions of convex polytopes
凸多面体的边缘度量尺寸
IF 1
3区 数学
Meiqin Wei , Bohua Fan , Changhong Lu , Jun Yue , Jinfeng Liu
{"title":"The edge metric dimensions of convex polytopes","authors":"Meiqin Wei , Bohua Fan , Changhong Lu , Jun Yue , Jinfeng Liu","doi":"10.1016/j.dam.2025.07.027","DOIUrl":"10.1016/j.dam.2025.07.027","url":null,"abstract":"<div><div>Let <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> be a connected graph. A vertex <span><math><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow></math></span> distinguishes the edge pair <span><math><mrow><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><mo>∈</mo><mi>E</mi></mrow></math></span> if the distances from <span><math><mi>x</mi></math></span> to <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> are distinct. A vertex subset <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is an edge metric generator of <span><math><mi>G</mi></math></span> if any pair of edges in <span><math><mi>E</mi></math></span> can be distinguished by some element of <span><math><mi>S</mi></math></span>. The minimum size of an edge metric generator of <span><math><mi>G</mi></math></span> is called the edge metric dimension of <span><math><mi>G</mi></math></span> and denoted by <span><math><mrow><mi>e</mi><mi>d</mi><mi>i</mi><mi>m</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we determine the exact values of the edge metric dimensions for some convex polytopes and generalized convex polytopes, which further emphasize the fact that there are families of convex polytopes having greater edge metric dimensions than their metric dimensions. The proof methods in this paper are constructive and they can be implemented through algorithms.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 294-306"},"PeriodicalIF":1.0,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724344","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}
引用次数: 0
Perfect k-matching, k-factor-critical and Aα-spectral radius
完美的k匹配，k因子临界和a α谱半径
IF 1
3区 数学
Mengyuan Niu , Shanshan Zhang , Xiumei Wang
{"title":"Perfect k-matching, k-factor-critical and Aα-spectral radius","authors":"Mengyuan Niu , Shanshan Zhang , Xiumei Wang","doi":"10.1016/j.dam.2025.07.020","DOIUrl":"10.1016/j.dam.2025.07.020","url":null,"abstract":"<div><div>A <span><math><mi>k</mi></math></span>-<em>matching</em> of a graph <span><math><mi>G</mi></math></span> is a function <span><math><mi>f</mi></math></span>: <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> satisfying <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>∂</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></munder><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≤</mo><mi>k</mi></mrow></math></span> for any vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. A <span><math><mi>k</mi></math></span>-matching of a graph <span><math><mi>G</mi></math></span> is <em>perfect</em> if <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>∂</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></munder><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span> for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. A graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> is <em>k-factor-critical</em> if the removal of any set of <span><math><mi>k</mi></math></span> vertices of <span><math><mi>G</mi></math></span> results in a graph with a perfect matching. Let <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the adjacency matrix and the degree diagonal matrix of <span><math><mi>G</mi></math></span>. For <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, Nikiforov <span><math><mrow><mo>(</mo><mn>2017</mn><mo>)</mo></mrow></math></span> introduced the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrix of G as follows: <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> In this paper, according to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius, we provide two sufficient conditions to ensure that a graph is <span><math><mi>k</mi></math></span>-factor-critical and has a perfect <span><math><mi>k</mi></math></span>-matching, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 384-393"},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721096","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}
引用次数: 0
Sequential testing problem: A follow-up review
顺序测试问题：后续审查
IF 1
3区 数学
Tonguç Ünlüyurt
{"title":"Sequential testing problem: A follow-up review","authors":"Tonguç Ünlüyurt","doi":"10.1016/j.dam.2025.07.039","DOIUrl":"10.1016/j.dam.2025.07.039","url":null,"abstract":"<div><div>This review aims to provide a comprehensive update on the progress made on the Sequential Testing problem (STP) in the last 20 years after the review, Ünlüyurt (2004) was published. Many studies have provided new theoretical results, extensions of the problem, and new applications. In this review, we pinpoint the main results and discuss the relations between the problems studied. We also provide possible research directions for the problem.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 356-369"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713417","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}
引用次数: 0
About the infinite windy firebreak location problem
关于无限风防火墙的选址问题
IF 1
3区 数学
Marc Demange , Alessia Di Fonso , Gabriele Di Stefano , Pierpaolo Vittorini
{"title":"About the infinite windy firebreak location problem","authors":"Marc Demange , Alessia Di Fonso , Gabriele Di Stefano , Pierpaolo Vittorini","doi":"10.1016/j.dam.2025.07.028","DOIUrl":"10.1016/j.dam.2025.07.028","url":null,"abstract":"<div><div>The severity of wildfires can be mitigated using preventive measures like the construction of firebreaks, which are strips of land from which the vegetation is completely removed. In this paper, we model the problem of wildfire containment as an optimization problem on infinite graphs called <span>Infinite Windy Firebreak Location</span>. A land of unknown size is modeled as an infinite undirected graph in which the vertices correspond to areas subject to fire and edges represent fire propagation from one area to another. A firebreak construction is modeled as removing the edge between two vertices. The number of firebreaks that can be installed depends on budget constraints. We assume that a fire ignites in a subset of vertices and propagates to the neighbors. The goal is to select a subset of edges to remove in order to contain the fire and avoid burning an infinite part of the graph. We prove that <span>Infinite Windy Firebreak Location</span> is coNP-complete in restricted cases, and we address some polynomial cases. We show that <span>Infinite Windy Firebreak Location</span> polynomially reduces to <span>Min Cut</span> for certain classes of graphs like infinite grid graphs and polyomino-grids.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 280-293"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144714080","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}
引用次数: 0
A note on obtaining bipartite radio graceful graphs of arbitrarily large radio numbers with radio graceful complements
关于获得具有任意大无线电数的无线电优美补的二部无线电优美图的注记
IF 1
3区 数学
Ushnish Sarkar
{"title":"A note on obtaining bipartite radio graceful graphs of arbitrarily large radio numbers with radio graceful complements","authors":"Ushnish Sarkar","doi":"10.1016/j.dam.2025.07.012","DOIUrl":"10.1016/j.dam.2025.07.012","url":null,"abstract":"<div><div>Motivated by the frequency assignment problem (FAP), a radio coloring of a graph <span><math><mi>G</mi></math></span> is an assignment <span><math><mi>f</mi></math></span> of non-negative integers to the vertices of <span><math><mi>G</mi></math></span> satisfying the condition <span><math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>+</mo><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>≥</mo><mtext>diameter of</mtext><mspace></mspace><mi>G</mi><mo>+</mo><mn>1</mn></mrow></math></span>, where <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> is the distance between any two vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> of the graph <span><math><mi>G</mi></math></span>. The span of a radio coloring of <span><math><mi>G</mi></math></span> is the difference of the maximum and minimum non-negative integers used as colors. The minimum span of a radio coloring of <span><math><mi>G</mi></math></span> is referred as the radio number of <span><math><mi>G</mi></math></span>. Any radio coloring with the minimum span is referred as an optimal radio coloring of <span><math><mi>G</mi></math></span>. If an optimal radio coloring of <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 a bijection from <span><math><mi>V</mi></math></span> to <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>−</mo><mn>1</mn><mo>}</mo></mrow></math></span>, then the graph is referred as radio graceful. In this article, using a recursive construction, we have shown that for each positive integer <span><math><mrow><mi>n</mi><mo>≥</mo><mn>9</mn></mrow></math></span>, there exists a bipartite graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices such that both <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and its complement <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> are radio graceful graphs. In the process, we show that each such <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> contain a Hamiltonian path.</div><div>Note that our construction obtains radio graceful graphs of arbitrarily large radio numbers without going for big cliques. This has an interesting similarity with the motivation behind the Mycielski’s construction which ensures the exis","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 350-355"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144712954","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}
引用次数: 0
Some novel minimax results for perfect matchings of polyomino graphs
多项式图完美匹配的一些新颖的极大极小结果
IF 1
3区 数学
Chunhu Sun , Heping Zhang
{"title":"Some novel minimax results for perfect matchings of polyomino graphs","authors":"Chunhu Sun , Heping Zhang","doi":"10.1016/j.dam.2025.07.030","DOIUrl":"10.1016/j.dam.2025.07.030","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with a perfect matching <span><math><mi>M</mi></math></span>. The forcing number of <span><math><mi>M</mi></math></span> in <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span>, is the minimal size of an edge subset of <span><math><mi>M</mi></math></span> that are contained in no other perfect matchings of <span><math><mi>G</mi></math></span>, and the anti-forcing number of <span><math><mi>M</mi></math></span> in <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>a</mi><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span>, is the minimal size of an edge subset of <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> whose deletion results in a subgraph with a unique perfect matching <span><math><mi>M</mi></math></span>. For a polyomino graph <span><math><mi>P</mi></math></span>, Zhou and Zhang (2016) established a minimax result: For every perfect matching <span><math><mi>M</mi></math></span> of <span><math><mi>P</mi></math></span> with the maximum forcing number or minus one, <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> is equal to the maximum number of disjoint <span><math><mi>M</mi></math></span>-alternating squares in <span><math><mi>P</mi></math></span>. In this paper, we show that for every perfect matching <span><math><mi>M</mi></math></span> of a polyomino graph <span><math><mi>P</mi></math></span> which contains no 3 × 3 chessboard as a nice subgraph, <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> is equal to the maximum number of disjoint <span><math><mi>M</mi></math></span>-alternating squares in <span><math><mi>P</mi></math></span>. Further we show that for every perfect matching <span><math><mi>M</mi></math></span> of <span><math><mi>P</mi></math></span>, <span><math><mrow><mi>a</mi><mi>f</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> always equals the number of <span><math><mi>M</mi></math></span>-alternating squares of <span><math><mi>P</mi></math></span> if and only if <span><math><mi>P</mi></math></span> has no 1 × 3 chessboard as a nice subgraph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 270-279"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144714079","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}
引用次数: 0
On z-coloring and b∗-coloring of graphs as improved variants of the b-coloring
图的z-着色和b * -着色作为b-着色的改进变体
IF 1
3区 数学
Manouchehr Zaker
{"title":"On z-coloring and b∗-coloring of graphs as improved variants of the b-coloring","authors":"Manouchehr Zaker","doi":"10.1016/j.dam.2025.07.036","DOIUrl":"10.1016/j.dam.2025.07.036","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a simple graph and <span><math><mi>c</mi></math></span> a proper vertex coloring of <span><math><mi>G</mi></math></span>. A vertex <span><math><mi>u</mi></math></span> is called b-vertex in <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></math></span> if all colors except <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> appear in the neighborhood of <span><math><mi>u</mi></math></span>. By a <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-coloring of <span><math><mi>G</mi></math></span> using colors <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></math></span> we define a proper vertex coloring <span><math><mi>c</mi></math></span> such that there is a b-vertex <span><math><mi>u</mi></math></span> (called nice vertex) such that <span><math><mi>u</mi></math></span> is adjacent to a b-vertex of color <span><math><mi>j</mi></math></span> for each <span><math><mrow><mi>j</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> with <span><math><mrow><mi>j</mi><mo>≠</mo><mi>c</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>. The <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-chromatic number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the largest integer <span><math><mi>k</mi></math></span> such that <span><math><mi>G</mi></math></span> has a <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-coloring using <span><math><mi>k</mi></math></span> colors. Every graph <span><math><mi>G</mi></math></span> admits a <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-coloring which is an improvement over the famous b-coloring. A z-coloring of <span><math><mi>G</mi></math></span> is a coloring <span><math><mi>c</mi></math></span> using colors <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></math></span> containing a nice vertex of color <span><math><mi>k</mi></math></span> such that for each two colors <span><math><mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></math></span>, each vertex of color <span><math><mi>j</mi></math></span> has a neighbor of color <span><math><mi>i</mi></math></span> in the graph (i.e. <span><math><mi>c</mi></math></span> is obtained from a Grundy-coloring of <span><math><mi>G</mi></math></span>). We prove that <span><math><mrow><msup><mrow><mi>b</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> cannot be approximated within any constant factor unless <span><math><mrow><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 370-379"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713418","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}
引用次数: 0
The thickness of a graph with constraint on girth
有周长约束的图的厚度
IF 1
3区 数学
Dengju Ma
{"title":"The thickness of a graph with constraint on girth","authors":"Dengju Ma","doi":"10.1016/j.dam.2025.07.025","DOIUrl":"10.1016/j.dam.2025.07.025","url":null,"abstract":"<div><div>The thickness of a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum integer <span><math><mi>t</mi></math></span> such that <span><math><mi>G</mi></math></span> can be decomposed into <span><math><mi>t</mi></math></span> planar subgraphs. In 1991, Dean et al. conjectured that the thickness of a graph <span><math><mi>G</mi></math></span> with <span><math><mi>m</mi></math></span> edges is at most <span><math><mrow><msqrt><mrow><mi>m</mi><mo>/</mo><mn>16</mn></mrow></msqrt><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. In this paper we show that the conjecture holds if the girth <span><math><mi>g</mi></math></span> of <span><math><mi>G</mi></math></span> is at least five, and that <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo><</mo><mn>1</mn><mo>+</mo><mroot><mrow><mi>m</mi><mo>/</mo><mn>2</mn></mrow><mrow><mi>s</mi></mrow></mroot></mrow></math></span> if <span><math><mrow><mi>g</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, where <span><math><mrow><mi>s</mi><mo>=</mo><mrow><mo>⌊</mo><mi>g</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></mrow></math></span>. If the girth of <span><math><mi>G</mi></math></span> is not restricted, then we prove that <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><mn>5</mn><mo>/</mo><mn>6</mn><mo>+</mo><msqrt><mrow><mn>2</mn><mi>m</mi><mo>/</mo><mn>9</mn><mo>−</mo><mn>23</mn><mo>/</mo><mn>36</mn></mrow></msqrt><mo>⌋</mo></mrow></mrow></math></span> where <span><math><mrow><mi>m</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, which improves the known bound for the thickness of graphs. If <span><math><mi>G</mi></math></span> is a connected nonplanar graph with girth <span><math><mrow><mi>g</mi><mo>≥</mo><mn>6</mn></mrow></math></span> which is 2-cell embedded in the orientable surface <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> (the nonorientable surface <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>, respectively), then <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn><mo>+</mo><mrow><mo>⌊</mo><mroot><mrow><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mroot><mo>⌋</mo></mrow></mrow></math></span>\u0000 (<span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn><mo>+</mo><mrow><mo>⌊</mo><mroot><mrow><mi>h</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mroot><mo>⌋</mo></mrow></mrow></math></span>,respectively). Clearly, if <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> or <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> is fixed, then the aforementioned bounds get closer and closer to 4, as the girt","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 374-383"},"PeriodicalIF":1.0,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144704422","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}
引用次数: 0
Intersection of cycles and paths in k-connected graphs
k连通图中环与路径的交
IF 1
3区 数学
Haidong Wu
{"title":"Intersection of cycles and paths in k-connected graphs","authors":"Haidong Wu","doi":"10.1016/j.dam.2025.07.013","DOIUrl":"10.1016/j.dam.2025.07.013","url":null,"abstract":"<div><div>McGuinness (2005) shows that if <span><math><mi>G</mi></math></span> is a <span><math><mi>k</mi></math></span>-connected graph <span><math><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></math></span> having circumference <span><math><mrow><mi>c</mi><mo>=</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mi>k</mi></mrow></math></span>, then for a pair of cycles <span><math><mi>C</mi></math></span> and <span><math><mi>D</mi></math></span> of <span><math><mi>G</mi></math></span> such that <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mn>2</mn><mi>c</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></math></span>, it must be true that <span><math><mi>C</mi></math></span> and <span><math><mi>D</mi></math></span> intersect in at least two common vertices. Using this result, McGuinness proves that for any <span><math><mi>k</mi></math></span>-connected graph <span><math><mi>G</mi></math></span> where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and having circumference <span><math><mrow><mi>c</mi><mo>≥</mo><mn>2</mn><mi>k</mi></mrow></math></span>, there is a bond <span><math><mi>B</mi></math></span> which intersects every cycle of length <span><math><mrow><mi>c</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>2</mn></mrow></math></span> or greater.</div><div>In this paper, we study the following general questions: will two long cycles or two paths intersect at a large number of vertices in a highly connected graph? We give positive answers to both questions and extend McGuinness’ result. Let <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the circumference of a graph <span><math><mi>G</mi></math></span>. We prove the following results.</div><div>(1) Let <span><math><mi>G</mi></math></span> be a <span><math><mi>k</mi></math></span>-connected graph for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>13</mn><mo>.</mo><mn>9413</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. Suppose <span><math><mi>C</mi></math></span> and <span><math><mi>D</mi></math></span> are cycles of <span><math><mi>G</mi></math></span> such that <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mn>2</mn><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><mi>c</mi><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow><mrow><mi>s</mi></mrow></mfrac><mo>+</mo><mn>12</mn></mrow></math></span> where <span><math><mrow><mi>c</mi><mo>=</mo><mn>2</mn><msqrt><mrow><mfrac><mrow><mn>13</mn></mrow><mrow><mn>7</mn></mrow></mfrac>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 226-233"},"PeriodicalIF":1.0,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703913","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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