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Five‐cycle double cover and shortest cycle cover 五周期双覆盖和最短周期覆盖
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-08-12 DOI: 10.1002/jgt.23164
Siyan Liu, Rong‐Xia Hao, Rong Luo, Cun‐Quan Zhang
{"title":"Five‐cycle double cover and shortest cycle cover","authors":"Siyan Liu, Rong‐Xia Hao, Rong Luo, Cun‐Quan Zhang","doi":"10.1002/jgt.23164","DOIUrl":"https://doi.org/10.1002/jgt.23164","url":null,"abstract":"The 5‐even subgraph cycle double cover conjecture (5‐CDC conjecture) asserts that every bridgeless graph has a 5‐even subgraph double cover. A shortest even subgraph cover of a graph is a family of even subgraphs which cover all the edges of and the sum of their lengths is minimum. It is conjectured that every bridgeless graph has an even subgraph cover with total length at most . In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5‐CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph satisfying the sufficient condition has a 4‐even subgraph ‐cover with total length at most . We also show that every oddness 2 cubic graph with girth at least 30 has a 5‐CDC containing a member of length at least and thus it has a 4‐even subgraph ‐cover with total length at most .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943300","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
Dense circuit graphs and the planar Turán number of a cycle 密集电路图和循环的平面图兰数
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-08-08 DOI: 10.1002/jgt.23165
Ruilin Shi, Zach Walsh, Xingxing Yu
{"title":"Dense circuit graphs and the planar Turán number of a cycle","authors":"Ruilin Shi, Zach Walsh, Xingxing Yu","doi":"10.1002/jgt.23165","DOIUrl":"https://doi.org/10.1002/jgt.23165","url":null,"abstract":"The <jats:italic>planar Turán number</jats:italic> of a graph is the maximum number of edges in an ‐vertex planar graph without as a subgraph. Let denote the cycle of length . The planar Turán number is known for . We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant so that for all and . When this bound is tight up to the constant and proves a conjecture of Cranston, Lidický, Liu, and Shantanam.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943337","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
Strong arc decompositions of split digraphs 分裂图的强弧分解
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-08-07 DOI: 10.1002/jgt.23157
Jørgen Bang‐Jensen, Yun Wang
{"title":"Strong arc decompositions of split digraphs","authors":"Jørgen Bang‐Jensen, Yun Wang","doi":"10.1002/jgt.23157","DOIUrl":"https://doi.org/10.1002/jgt.23157","url":null,"abstract":"A <jats:italic>strong arc decomposition</jats:italic> of a digraph is a partition of its arc set into two sets such that the digraph is strong for . Bang‐Jensen and Yeo conjectured that there is some such that every ‐arc‐strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2‐arc‐strong semicomplete digraph has a strong arc decomposition. Bang‐Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2‐arc‐strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3‐arc‐strong locally semicomplete digraph has a strong arc decomposition. A <jats:italic>split digraph</jats:italic> is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph by adding a new set of vertices and some arcs between and . In this paper, we prove that every 3‐arc‐strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2‐strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943301","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
Stability of Rose Window graphs 玫瑰窗图形的稳定性
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-08-06 DOI: 10.1002/jgt.23162
Milad Ahanjideh, István Kovács, Klavdija Kutnar
{"title":"Stability of Rose Window graphs","authors":"Milad Ahanjideh, István Kovács, Klavdija Kutnar","doi":"10.1002/jgt.23162","DOIUrl":"https://doi.org/10.1002/jgt.23162","url":null,"abstract":"A graph is said to be stable if for the direct product is isomorphic to ; otherwise, it is called unstable. An unstable graph is called nontrivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all nontrivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943338","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
Proper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s 具有彩虹 C4 ${C}_{4}$-s 的平面图的适当边着色
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-08-06 DOI: 10.1002/jgt.23163
András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy
{"title":"Proper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s","authors":"András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy","doi":"10.1002/jgt.23163","DOIUrl":"https://doi.org/10.1002/jgt.23163","url":null,"abstract":"We call a proper edge coloring of a graph a B‐coloring if every 4‐cycle of is colored with four different colors. Let denote the smallest number of colors needed for a B‐coloring of . Motivated by earlier papers on B‐colorings, here we consider for planar and outerplanar graphs in terms of the maximum degree . We prove that for planar graphs, for bipartite planar graphs, and for outerplanar graphs with . We conjecture that, for sufficiently large, for planar , and for outerplanar .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943351","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
Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs” 对 "在随机几何图中嵌入平衡生成树的锐阈值 "的更正
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-07-31 DOI: 10.1002/jgt.23160
{"title":"Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”","authors":"","doi":"10.1002/jgt.23160","DOIUrl":"https://doi.org/10.1002/jgt.23160","url":null,"abstract":"","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866312","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 the minimum number of arcs in 4‐dicritical oriented graphs 论四临界定向图中的最小弧数
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-07-30 DOI: 10.1002/jgt.23159
Frédéric Havet, Lucas Picasarri‐Arrieta, Clément Rambaud
{"title":"On the minimum number of arcs in 4‐dicritical oriented graphs","authors":"Frédéric Havet, Lucas Picasarri‐Arrieta, Clément Rambaud","doi":"10.1002/jgt.23159","DOIUrl":"https://doi.org/10.1002/jgt.23159","url":null,"abstract":"The dichromatic number of a digraph is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph is ‐dicritical if and each proper subdigraph of satisfies . For integers and , we define (resp., ) as the minimum number of arcs possible in a ‐dicritical digraph (resp., oriented graph). Kostochka and Stiebitz have shown that . They also conjectured that there is a constant such that for and large enough. This conjecture is known to be true for . In this work, we prove that every 4‐dicritical oriented graph on vertices has at least arcs, showing the conjecture for . We also characterise exactly the 4‐dicritical digraphs on vertices with exactly arcs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866313","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
Counting triangles in regular graphs 计算规则图形中的三角形
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-07-26 DOI: 10.1002/jgt.23156
Jialin He, Xinmin Hou, Jie Ma, Tianying Xie
{"title":"Counting triangles in regular graphs","authors":"Jialin He, Xinmin Hou, Jie Ma, Tianying Xie","doi":"10.1002/jgt.23156","DOIUrl":"https://doi.org/10.1002/jgt.23156","url":null,"abstract":"In this paper, we investigate the minimum number of triangles, denoted by , in ‐vertex ‐regular graphs, where is an odd integer and is an even integer. The well‐known Andrásfai–Erdős–Sós Theorem has established that if . In a striking work, Lo has provided the exact value of for sufficiently large , given that . Here, we bridge the gap between the aforementioned results by determining the precise value of in the entire range . This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771226","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
Counting rainbow triangles in edge‐colored graphs 计算边色图中的彩虹三角形
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-07-24 DOI: 10.1002/jgt.23158
Xueliang Li, Bo Ning, Yongtang Shi, Shenggui Zhang
{"title":"Counting rainbow triangles in edge‐colored graphs","authors":"Xueliang Li, Bo Ning, Yongtang Shi, Shenggui Zhang","doi":"10.1002/jgt.23158","DOIUrl":"https://doi.org/10.1002/jgt.23158","url":null,"abstract":"Let be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . As a consequence, we prove counting results for rainbow triangles in edge‐colored graphs. One main theorem states that the number of rainbow triangles in is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph (i.e., rainbow triangles sharing a common vertex).","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771227","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
Defective acyclic colorings of planar graphs 平面图形的缺陷非循环着色
IF 0.9 3区 数学
Journal of Graph Theory Pub Date : 2024-07-22 DOI: 10.1002/jgt.23154
On‐Hei Solomon Lo, Ben Seamone, Xuding Zhu
{"title":"Defective acyclic colorings of planar graphs","authors":"On‐Hei Solomon Lo, Ben Seamone, Xuding Zhu","doi":"10.1002/jgt.23154","DOIUrl":"https://doi.org/10.1002/jgt.23154","url":null,"abstract":"This paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744614","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
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