Pub Date : 2024-07-26DOI: 10.1016/j.disc.2024.114161
Given a set of graphs, we call a copy of a graph in an -graph. The -isolation number of a graph G, denoted by , is the size of a smallest subset D of the vertex set such that the closed neighbourhood of D intersects the vertex sets of the -graphs contained by G (equivalently, contains no -graph). Thus, is the domination number of G. The second author showed that if is the set of cycles and G is a connected n-vertex graph that is not a triangle, then . This bound is attainable for every n and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if for some , where is a cycle of length k. The problem is to determine the smallest real number (if it exists) such that for some finite set of graphs, for every connected graph G that is not an -graph. The above-mentioned result yields and . The second author also showed that if
给定一个图集 F,我们称 F 中一个图的副本为 F 图。图 G 的 F 隔离数用 ι(G,F)表示,是顶点集 V(G)的最小子集 D 的大小,D 的封闭邻域与 G 所包含的 F 图的顶点集相交(等价地,G-N[D] 不包含任何 F 图)。因此,ι(G,{K1}) 是 G 的支配数。第二位作者证明,如果 F 是循环集,而 G 是一个非三角形的 n 顶点连通图,那么 ι(G,F)≤⌊n4⌋。对于每一个 n,这个界限都是可以达到的,并且解决了卡罗和汉斯伯格的一个问题。随即产生的一个问题是,如果 F={Ck} 为某个 k≥3,其中 Ck 是长度为 k 的循环,那么上限还能小多少?问题是确定最小实数 ck(如果存在),使得对于某个有限图集 Ek,ι(G,{Ck})≤ck|V(G)| 对于每个非 Ek 图的连通图 G,ι(G,{Ck})≤ck|V(G)|。根据上述结果可以得出 c3=14 和 E3={C3}。第二位作者还证明,如果 k≥5 且 ck 存在,则 ck≥22k+1。我们证明 c4=15 并确定 E4,它由三个 4 顶点图和六个 9 顶点图组成。E4 中的 9 顶点图是通过计算机程序完全确定的。本文介绍了一种有可能得出类似结果的方法。
{"title":"Isolation of squares in graphs","authors":"","doi":"10.1016/j.disc.2024.114161","DOIUrl":"10.1016/j.disc.2024.114161","url":null,"abstract":"<div><p>Given a set <span><math><mi>F</mi></math></span> of graphs, we call a copy of a graph in <span><math><mi>F</mi></math></span> an <span><math><mi>F</mi></math></span>-graph. The <span><math><mi>F</mi></math></span>-isolation number of a graph <em>G</em>, denoted by <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the size of a smallest subset <em>D</em> of the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that the closed neighbourhood of <em>D</em> intersects the vertex sets of the <span><math><mi>F</mi></math></span>-graphs contained by <em>G</em> (equivalently, <span><math><mi>G</mi><mo>−</mo><mi>N</mi><mo>[</mo><mi>D</mi><mo>]</mo></math></span> contains no <span><math><mi>F</mi></math></span>-graph). Thus, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is the domination number of <em>G</em>. The second author showed that if <span><math><mi>F</mi></math></span> is the set of cycles and <em>G</em> is a connected <em>n</em>-vertex graph that is not a triangle, then <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></math></span>. This bound is attainable for every <em>n</em> and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for some <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a cycle of length <em>k</em>. The problem is to determine the smallest real number <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> (if it exists) such that for some finite set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of graphs, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> for every connected graph <em>G</em> that is not an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-graph. The above-mentioned result yields <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>. The second author also showed that if <span><math><mi>k</mi><mo","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953076","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}
Pub Date : 2024-07-26DOI: 10.1016/j.disc.2024.114181
A pair of hypergraphs is called orthogonal if for every pair of edges . An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gyárfás and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gyárfás–Lehel conjecture.
{"title":"Looms","authors":"","doi":"10.1016/j.disc.2024.114181","DOIUrl":"10.1016/j.disc.2024.114181","url":null,"abstract":"<div><p>A pair <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> of hypergraphs is called <em>orthogonal</em> if <span><math><mo>|</mo><mi>a</mi><mo>∩</mo><mi>b</mi><mo>|</mo><mo>=</mo><mn>1</mn></math></span> for every pair of edges <span><math><mi>a</mi><mo>∈</mo><mi>A</mi><mo>,</mo><mspace></mspace><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>. An orthogonal pair of hypergraphs is called a <em>loom</em> if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gyárfás and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gyárfás–Lehel conjecture.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953077","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}
Pub Date : 2024-07-26DOI: 10.1016/j.disc.2024.114184
Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting (AEAQEC, for short) code, and gave some good AEAQEC codes. In this paper, we provide two methods of constructing AEAQEC codes by means of two matrix-product codes from constacyclic codes over finite fields. The first one is derived from the rank of a relationship of generator matrices based on two matrix-product codes. The second construction is derived from the dimension of intersection for two matrix-product codes. By means of these methods, concrete examples are presented to construct new AEAQEC codes. In addition, our obtained AEQAEC codes have better parameters than the ones available in the literature.
{"title":"Constructions of AEAQEC codes via matrix-product codes","authors":"","doi":"10.1016/j.disc.2024.114184","DOIUrl":"10.1016/j.disc.2024.114184","url":null,"abstract":"<div><p>Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting (AEAQEC, for short) code, and gave some good AEAQEC codes. In this paper, we provide two methods of constructing AEAQEC codes by means of two matrix-product codes from constacyclic codes over finite fields. The first one is derived from the rank of a relationship of generator matrices based on two matrix-product codes. The second construction is derived from the dimension of intersection for two matrix-product codes. By means of these methods, concrete examples are presented to construct new AEAQEC codes. In addition, our obtained AEQAEC codes have better parameters than the ones available in the literature.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953083","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}
Pub Date : 2024-07-25DOI: 10.1016/j.disc.2024.114180
A graph is half-arc-transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is half-arc-regular if its full automorphism group acts regularly on its edges. A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called bi-dihedrant. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in [8, Theorem 6.7] completes the classification of edge-regular bi-dihedrants of valency 6.
{"title":"On hexavalent half-arc-transitive bi-dihedrants","authors":"","doi":"10.1016/j.disc.2024.114180","DOIUrl":"10.1016/j.disc.2024.114180","url":null,"abstract":"<div><p>A graph is <em>half-arc-transitive</em> if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is <em>half-arc-regular</em> if its full automorphism group acts regularly on its edges. A graph is said to be a <em>bi-Cayley graph</em> over a group <em>H</em> if it admits <em>H</em> as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called <em>bi-dihedrant</em>. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in <span><span>[8, Theorem 6.7]</span></span> completes the classification of edge-regular bi-dihedrants of valency 6.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933432","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}
Pub Date : 2024-07-24DOI: 10.1016/j.disc.2024.114177
The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by linking Caylerian polynomials to Burge words and Burge matrices. The γ-nonnegativity of the two-sided Eulerian polynomials is reformulated in terms of Burge structures. Finally, Cayley permutations with a prescribed ascent set are shown to be counted by Burge matrices with fixed row sums.
{"title":"Caylerian polynomials","authors":"","doi":"10.1016/j.disc.2024.114177","DOIUrl":"10.1016/j.disc.2024.114177","url":null,"abstract":"<div><p>The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by linking Caylerian polynomials to Burge words and Burge matrices. The <em>γ</em>-nonnegativity of the two-sided Eulerian polynomials is reformulated in terms of Burge structures. Finally, Cayley permutations with a prescribed ascent set are shown to be counted by Burge matrices with fixed row sums.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951780","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}
Pub Date : 2024-07-24DOI: 10.1016/j.disc.2024.114173
The edge-face chromatic number of a plane graph G is the least number of colors such that any two adjacent or incident elements in receive different colors. In 2005, Luo and Zhang proved that each 2-connected simple graph G with has . The condition is improved to in this paper.
平面图 G 的边-面色度数 χef(G)是 E(G)∪F(G) 中任意两个相邻或入射元素获得不同颜色的最少颜色数。2005 年,Luo 和 Zhang 证明了每个 Δ≥24 的 2 连接简单图 G 都有χef(G)=Δ。本文将条件 Δ⩾24 改进为 Δ⩾13。
{"title":"An improved upper bound on the edge-face coloring of 2-connected plane graphs","authors":"","doi":"10.1016/j.disc.2024.114173","DOIUrl":"10.1016/j.disc.2024.114173","url":null,"abstract":"<div><p>The edge-face chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>e</mi><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a plane graph <em>G</em> is the least number of colors such that any two adjacent or incident elements in <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∪</mo><mi>F</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> receive different colors. In 2005, Luo and Zhang proved that each 2-connected simple graph <em>G</em> with <span><math><mi>Δ</mi><mo>≥</mo><mn>24</mn></math></span> has <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>e</mi><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi></math></span>. The condition <span><math><mi>Δ</mi><mo>⩾</mo><mn>24</mn></math></span> is improved to <span><math><mi>Δ</mi><mo>⩾</mo><mn>13</mn></math></span> in this paper.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141960871","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}
Pub Date : 2024-07-24DOI: 10.1016/j.disc.2024.114176
We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct neighborhoods within S. The smallest size of such a set is denoted by . It has been conjectured that holds for every twin-free graph G of order n without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.
我们研究图中最优定位总支配集大小的上限。如果图 G 的每个顶点在 S 中都有一个邻居,并且 S 外的任何两个顶点在 S 中都有不同的邻域,那么图 G 的顶点集合 S 就是定位-总支配集。有人猜想,γtL(G)≤2n3 对于每一个无孤立顶点的 n 阶无孪生图 G 都成立。我们证明该猜想对于共方图、分裂图、块图和次立方图都成立。
{"title":"Progress towards the two-thirds conjecture on locating-total dominating sets","authors":"","doi":"10.1016/j.disc.2024.114176","DOIUrl":"10.1016/j.disc.2024.114176","url":null,"abstract":"<div><p>We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set <em>S</em> of vertices of a graph <em>G</em> is a locating-total dominating set if every vertex of <em>G</em> has a neighbor in <em>S</em>, and if any two vertices outside <em>S</em> have distinct neighborhoods within <em>S</em>. The smallest size of such a set is denoted by <span><math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It has been conjectured that <span><math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> holds for every twin-free graph <em>G</em> of order <em>n</em> without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141960872","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}
Pub Date : 2024-07-24DOI: 10.1016/j.disc.2024.114179
An association scheme is P-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate P-polynomial association schemes of type were introduced by Bernard et al., and multivariate P-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of m-distance-regular graph is defined and shown to give a graph interpretation of the multivariate P-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate P-polynomial association schemes and m-distance-regular graphs are also considered. In particular, bivariate P-polynomial schemes of type are discussed, and their connection to 2-distance-regular graphs is established.
当且仅当关联方案由距离规则图的距离矩阵组成时,它才是 P 多项式关联方案。最近,Bernard 等人提出了 (α,β)类型的双变量 P 多项式关联方案,Bannai 等人随后定义了多变量 P 多项式关联方案。本文定义了 m 距离规则图的概念,并展示了多变量 P 多项式关联方案的图解释。文中提供了各种实例。本文还考虑了多变量 P 多项式关联方案和 m 距离不规则图的细化结构和附加约束。特别是讨论了 (α,β) 类型的双变量 P 多项式方案,并建立了它们与 2-距离不规则图的联系。
{"title":"m-Distance-regular graphs and their relation to multivariate P-polynomial association schemes","authors":"","doi":"10.1016/j.disc.2024.114179","DOIUrl":"10.1016/j.disc.2024.114179","url":null,"abstract":"<div><p>An association scheme is <em>P</em>-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate <em>P</em>-polynomial association schemes of type <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> were introduced by Bernard et al., and multivariate <em>P</em>-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of <em>m</em>-distance-regular graph is defined and shown to give a graph interpretation of the multivariate <em>P</em>-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate <em>P</em>-polynomial association schemes and <em>m</em>-distance-regular graphs are also considered. In particular, bivariate <em>P</em>-polynomial schemes of type <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> are discussed, and their connection to 2-distance-regular graphs is established.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003108/pdfft?md5=432f38716b4f07d3d94b55cbb29bcfe3&pid=1-s2.0-S0012365X24003108-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141960873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-24DOI: 10.1016/j.disc.2024.114175
The Erdős-Gyárfás conjecture asserts that every graph with minimum degree at least three has a cycle whose length is a power of 2. Let G be a graph with minimum degree at least 3. We show that if G contains no induced path of order 10, then G contains a cycle of length 4 or 8, and hence the conjecture holds in this case.
厄尔多斯-希亚法斯猜想断言,每个最小度数至少为 3 的图都有一个长度为 2 的幂的循环。让 G 是一个最小度数至少为 3 的图。我们证明,如果 G 不包含阶数为 10 的诱导路径,那么 G 包含一个长度为 4 或 8 的循环,因此猜想在这种情况下成立。
{"title":"The Erdős-Gyárfás conjecture holds for P10-free graphs","authors":"","doi":"10.1016/j.disc.2024.114175","DOIUrl":"10.1016/j.disc.2024.114175","url":null,"abstract":"<div><p>The Erdős-Gyárfás conjecture asserts that every graph with minimum degree at least three has a cycle whose length is a power of 2. Let <em>G</em> be a graph with minimum degree at least 3. We show that if <em>G</em> contains no induced path of order 10, then <em>G</em> contains a cycle of length 4 or 8, and hence the conjecture holds in this case.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951781","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}
Pub Date : 2024-07-22DOI: 10.1016/j.disc.2024.114172
Let G be a connected graph with at least vertices that contains a perfect matching. Then G is if for each pair of disjoint matchings of size m and n, respectively, there exists a perfect matching F in G such that and . A graph G is 1-embeddable in a surface Σ if G can be drawn in Σ so that every edge of G crosses at most one other edge at a point. R.E.L. Aldred and M.D. Plummer [1], [2] investigated the properties for graphs embedded in the plane, torus, projective plane and Klein bottle. In this paper, we study the property for 1-embeddable graphs in surfaces with small genus. It is shown that no 1-embeddable graph in the plane or projective plane is and no 1-embeddable graph in the torus or Klein bottle is . As corollaries, no 1-embeddable graph in the plane or projective plane is 5-extendable and no 1-embeddable graph in the torus or Klein bottle is 6-extendable. Some examples show that such results are best possible.
{"title":"On restricted matching extension of 1-embeddable graphs in surfaces with small genus","authors":"","doi":"10.1016/j.disc.2024.114172","DOIUrl":"10.1016/j.disc.2024.114172","url":null,"abstract":"<div><p>Let <em>G</em> be a connected graph with at least <span><math><mn>2</mn><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> vertices that contains a perfect matching. Then <em>G</em> is <span><math><mi>E</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> if for each pair of disjoint matchings <span><math><mi>M</mi><mo>,</mo><mi>N</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of size <em>m</em> and <em>n</em>, respectively, there exists a perfect matching <em>F</em> in <em>G</em> such that <span><math><mi>M</mi><mo>⊆</mo><mi>F</mi></math></span> and <span><math><mi>F</mi><mo>∩</mo><mi>N</mi><mo>=</mo><mo>∅</mo></math></span>. A graph <em>G</em> is <em>1-embeddable</em> in a surface Σ if <em>G</em> can be drawn in Σ so that every edge of <em>G</em> crosses at most one other edge at a point. R.E.L. Aldred and M.D. Plummer <span><span>[1]</span></span>, <span><span>[2]</span></span> investigated the properties <span><math><mi>E</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for graphs embedded in the plane, torus, projective plane and Klein bottle. In this paper, we study the property <span><math><mi>E</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for 1-embeddable graphs in surfaces with small genus. It is shown that no 1-embeddable graph in the plane or projective plane is <span><math><mi>E</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and no 1-embeddable graph in the torus or Klein bottle is <span><math><mi>E</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. As corollaries, no 1-embeddable graph in the plane or projective plane is 5-extendable and no 1-embeddable graph in the torus or Klein bottle is 6-extendable. Some examples show that such results are best possible.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933340","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}