A graph $G$ with an even number of edges is called even-decomposable if there is a sequence $V(G)=V_0supset V_1supset dots supset V_k=emptyset$ such that for each $i$, $G[V_i]$ has an even number of edges and $V_isetminus~V_{i+1}$ is an independent set in $G$. The study of this property was initiated recently by Versteegen, motivated by connections to a Ramsey-type problem and questions about graph codes posed by Alon. Resolving a conjecture of Versteegen, we prove that all but an $e^{-Omega(n^2)}$ proportion of the $n$-vertex graphs with an even number of edges are even-decomposable. Moreover, answering one of his questions, we determine the order of magnitude of the smallest $p=p(n)$ for which the probability that the random graph $G(n,1-p)$ is even-decomposable (conditional on it having an even number of edges) is at least $1/2$. We also study the following closely related property. A graph is called even-degenerate if there is an ordering $v_1,v_2,dots,v_n$ of its vertices such that each $v_i$ has an even number of neighbours in the set ${v_{i+1},dots,v_n}$. We prove that all but an $e^{-Omega(n)}$ proportion of the $n$-vertex graphs with an even number of edges are even-degenerate, which is tight up to the implied constant.
{"title":"The probability that a random graph is even-decomposable","authors":"Oliver Janzer, Fredy Yip","doi":"arxiv-2409.11152","DOIUrl":"https://doi.org/arxiv-2409.11152","url":null,"abstract":"A graph $G$ with an even number of edges is called even-decomposable if there\u0000is a sequence $V(G)=V_0supset V_1supset dots supset V_k=emptyset$ such\u0000that for each $i$, $G[V_i]$ has an even number of edges and\u0000$V_isetminus~V_{i+1}$ is an independent set in $G$. The study of this property\u0000was initiated recently by Versteegen, motivated by connections to a Ramsey-type\u0000problem and questions about graph codes posed by Alon. Resolving a conjecture\u0000of Versteegen, we prove that all but an $e^{-Omega(n^2)}$ proportion of the\u0000$n$-vertex graphs with an even number of edges are even-decomposable. Moreover,\u0000answering one of his questions, we determine the order of magnitude of the\u0000smallest $p=p(n)$ for which the probability that the random graph $G(n,1-p)$ is\u0000even-decomposable (conditional on it having an even number of edges) is at\u0000least $1/2$. We also study the following closely related property. A graph is called\u0000even-degenerate if there is an ordering $v_1,v_2,dots,v_n$ of its vertices\u0000such that each $v_i$ has an even number of neighbours in the set\u0000${v_{i+1},dots,v_n}$. We prove that all but an $e^{-Omega(n)}$ proportion\u0000of the $n$-vertex graphs with an even number of edges are even-degenerate,\u0000which is tight up to the implied constant.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For $textbf{r}=(r_1,ldots,r_k)$, an $textbf{r}$-factorization of the complete $lambda$-fold $h$-uniform $n$-vertex hypergraph $lambda K_n^h$ is a partition of the edges of $lambda K_n^h$ into $F_1,ldots, F_k$ such that $F_j$ is $r_j$-regular and spanning for $1leq jleq k$. This paper shows that for $n>frac{m-1}{1-2^{frac{1}{1-h}}}+h-1$, a partial $textbf{r}$-factorization of $lambda K_m^h$ can be extended to an $textbf{r}$-factorization of $lambda K_n^h$ if and only if the obvious necessary conditions are satisfied.
{"title":"Embedding arbitrary edge-colorings of hypergraphs into regular colorings","authors":"Xiaomiao Wang, Tao Feng, Shixin Wang","doi":"arxiv-2409.10950","DOIUrl":"https://doi.org/arxiv-2409.10950","url":null,"abstract":"For $textbf{r}=(r_1,ldots,r_k)$, an $textbf{r}$-factorization of the\u0000complete $lambda$-fold $h$-uniform $n$-vertex hypergraph $lambda K_n^h$ is a\u0000partition of the edges of $lambda K_n^h$ into $F_1,ldots, F_k$ such that\u0000$F_j$ is $r_j$-regular and spanning for $1leq jleq k$. This paper shows that\u0000for $n>frac{m-1}{1-2^{frac{1}{1-h}}}+h-1$, a partial\u0000$textbf{r}$-factorization of $lambda K_m^h$ can be extended to an\u0000$textbf{r}$-factorization of $lambda K_n^h$ if and only if the obvious\u0000necessary conditions are satisfied.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1988, ErdH{o}s suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In this paper, we study a natural generalization of the question of ErdH{o}s in which we replace `triangle' with `clique of order $k$' for ${kge 3}$. We completely resolve this generalized question with the characterization of all extremal graphs. Motivated by applications in data science, we also study another generalization of the question of ErdH{o}s where every edge is required to be in at least $ell$ triangles for $ellge 2$ instead of only one triangle. We completely resolve this problem for $ell = 2$.
{"title":"Sparse graphs with local covering conditions on edges","authors":"Debsoumya Chakraborti, Amirali Madani, Anil Maheshwari, Babak Miraftab","doi":"arxiv-2409.11216","DOIUrl":"https://doi.org/arxiv-2409.11216","url":null,"abstract":"In 1988, ErdH{o}s suggested the question of minimizing the number of edges\u0000in a connected $n$-vertex graph where every edge is contained in a triangle.\u0000Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger\u0000form. In this paper, we study a natural generalization of the question of\u0000ErdH{o}s in which we replace `triangle' with `clique of order $k$' for ${kge\u00003}$. We completely resolve this generalized question with the characterization\u0000of all extremal graphs. Motivated by applications in data science, we also\u0000study another generalization of the question of ErdH{o}s where every edge is\u0000required to be in at least $ell$ triangles for $ellge 2$ instead of only one\u0000triangle. We completely resolve this problem for $ell = 2$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-M'elou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-M'elou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps.
{"title":"A Galois structure on the orbit of large steps walks in the quadrant","authors":"Pierre Bonnet, Charlotte Hardouin","doi":"arxiv-2409.11084","DOIUrl":"https://doi.org/arxiv-2409.11084","url":null,"abstract":"The enumeration of weighted walks in the quarter plane reduces to studying a\u0000functional equation with two catalytic variables. When the steps of the walk\u0000are small, Bousquet-M'elou and Mishna defined a group called the group of the\u0000walk which turned out to be crucial in the classification of the small steps\u0000models. In particular, its action on the catalytic variables provides a\u0000convenient set of changes of variables in the functional equation. This\u0000particular set called the orbit has been generalized to models with arbitrary\u0000large steps by Bostan, Bousquet-M'elou and Melczer (BBMM). However, the orbit\u0000had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which\u0000extends the notion of the group of the walk to models with large steps. As an\u0000application, we look into a general strategy to prove the algebraicity of\u0000models with small backwards steps, which uses the fundamental objects that are\u0000invariants and decoupling. The group action on the orbit allows us to develop a\u0000Galoisian approach to these two notions. Up to the knowledge of the finiteness\u0000of the orbit, this gives systematic procedures to test their existence and\u0000construct them. Our constructions lead to the first proofs of algebraicity of\u0000weighted models with large steps, proving in particular a conjecture of BBMM,\u0000and allowing to find new algebraic models with large steps.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are $X$-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We then show that such digraphs can be recognized from their connection sets.
{"title":"Recognizing bicoset digraphs which are $X$-joins and automorphism groups of bicoset digraphs","authors":"Rachel Barber, Ted Dobson, Gregory Robson","doi":"arxiv-2409.11092","DOIUrl":"https://doi.org/arxiv-2409.11092","url":null,"abstract":"We examine bicoset digraphs and their natural properties from the point of\u0000view of symmetry. We then consider connected bicoset digraphs that are\u0000$X$-joins with collections of empty graphs, and show that their automorphism\u0000groups can be obtained from their natural irreducible quotients. We then show\u0000that such digraphs can be recognized from their connection sets.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristina BallantineCollege of the Holy Cross, George BeckDalhousie University, Mircea MercaNational University of Science and Tehnology Politehnica Bucharest, Bruce SaganMichigan State University
Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre_2 is injective as a map on binary partitions of n. In the present work we derive a host of identities involving the sequences which count the number of parts of a given value in the image of pre_2. These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research.
设 e_k(x_1,...,x_l)是一个基本对称多项式，设 mu =(mu_1,...,mu_l) 是一个整数分部。定义 pre_k(mu)为分区，其各部分是求值 e_k(mu_1,...,mu_l)中的和。对这种分区的研究是由 Ballantine、Beck 和 Merca 发起的，他们证明了（除其他外）pre_2 作为 n 的二进制分区上的映射是可注入的。其中包括生成函数、明确表达式和前差公式。我们将其中的一些方法推广到 d-ary 分区，并探索与颜色分区的联系。我们的技术包括在有根分区上使用生成函数和双射。最后，我们列出了一些猜想和未来的研究方向。
{"title":"Elementary symmetric partitions","authors":"Cristina BallantineCollege of the Holy Cross, George BeckDalhousie University, Mircea MercaNational University of Science and Tehnology Politehnica Bucharest, Bruce SaganMichigan State University","doi":"arxiv-2409.11268","DOIUrl":"https://doi.org/arxiv-2409.11268","url":null,"abstract":"Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu =\u0000(mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition\u0000whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of\u0000such partitions was initiated by Ballantine, Beck, and Merca who showed (among\u0000other things) that pre_2 is injective as a map on binary partitions of n. In\u0000the present work we derive a host of identities involving the sequences which\u0000count the number of parts of a given value in the image of pre_2. These include\u0000generating functions, explicit expressions, and formulas for forward\u0000differences. We generalize some of these to d-ary partitions and explore\u0000connections with color partitions. Our techniques include the use of generating\u0000functions and bijections on rooted partitions. We end with a list of\u0000conjectures and a direction for future research.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment of $k$ colors to $E(G)$ such that for every edge $ein E(G)$, there is a color that is assigned to exactly one edge among the closed neighborhood of $e$. The smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the conflict-free chromatic index of $G$, denoted $chi'_{CF}(G)$. Dc{e}bski and Przybya{l}o showed that $2lechi'_{CF}(T)le 3$ for every tree $T$ of size at least two. In this paper, we present an algorithm to determine that the conflict-free chromatic index of a tree without 2-degree vertices is 2 or 3, in time $O(n^3)$. This partially answer a question raised by Dc{e}bski and Przybya{l}o.
{"title":"Conflict-free chromatic index of trees","authors":"Shanshan Guo, Ethan Y. H. Li, Luyi Li, Ping Li","doi":"arxiv-2409.10899","DOIUrl":"https://doi.org/arxiv-2409.10899","url":null,"abstract":"A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment\u0000of $k$ colors to $E(G)$ such that for every edge $ein E(G)$, there is a color\u0000that is assigned to exactly one edge among the closed neighborhood of $e$. The\u0000smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the\u0000conflict-free chromatic index of $G$, denoted $chi'_{CF}(G)$. Dc{e}bski and\u0000Przybya{l}o showed that $2lechi'_{CF}(T)le 3$ for every tree $T$ of size at\u0000least two. In this paper, we present an algorithm to determine that the\u0000conflict-free chromatic index of a tree without 2-degree vertices is 2 or 3, in\u0000time $O(n^3)$. This partially answer a question raised by Dc{e}bski and\u0000Przybya{l}o.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nina Chiarelli, Vesna Iršič, Marko Jakovac, William B. Kinnersley, Mirjana Mikalački
Motivated by the burning and cooling processes, the burning game is introduced. The game is played on a graph $G$ by the two players (Burner and Staller) that take turns selecting vertices of $G$ to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all vertices of $G$ as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph $G$ is the game burning number $b_g(G)$ if Burner makes the first move, and the Staller-start game burning number $b_g'(G)$ if Staller starts. In this paper, basic bounds on $b_g(G)$ are given and Continuation Principle is established. Graphs with small game burning numbers are characterized and Nordhaus-Gaddum type results are obtained. An analogue of the burning number conjecture for the burning game is considered and graph products are studied.
{"title":"Burning game","authors":"Nina Chiarelli, Vesna Iršič, Marko Jakovac, William B. Kinnersley, Mirjana Mikalački","doi":"arxiv-2409.11328","DOIUrl":"https://doi.org/arxiv-2409.11328","url":null,"abstract":"Motivated by the burning and cooling processes, the burning game is\u0000introduced. The game is played on a graph $G$ by the two players (Burner and\u0000Staller) that take turns selecting vertices of $G$ to burn; as in the burning\u0000process, burning vertices spread fire to unburned neighbors. Burner aims to\u0000burn all vertices of $G$ as quickly as possible, while Staller wants the\u0000process to last as long as possible. If both players play optimally, then the\u0000number of time steps needed to burn the whole graph $G$ is the game burning\u0000number $b_g(G)$ if Burner makes the first move, and the Staller-start game\u0000burning number $b_g'(G)$ if Staller starts. In this paper, basic bounds on\u0000$b_g(G)$ are given and Continuation Principle is established. Graphs with small\u0000game burning numbers are characterized and Nordhaus-Gaddum type results are\u0000obtained. An analogue of the burning number conjecture for the burning game is\u0000considered and graph products are studied.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog of facially complete graphs: they fall into seven types. A consequence is that if q is the size of the largest face in a plane graph G that is facially complete, then G has at most Floor[3/2 q] vertices. This bound was known, but our proof is completely different from the 1998 approach of Chen, Grigni, and Papadimitriou. Our method also yields a count of the 2-connected facially complete graphs with n vertices. We also show that if a plane graph has at most two faces of size 4 and no larger face, then the addition of both diagonals to each 4-face leads to a graph that is 5-colorable.
当地图中的区域有非空交点时，就认为它们是相邻的（而不是传统的要求在线段上有交点的观点），这就产生了面完全图的概念：当位于一个面上的每两个顶点之间都添加了边时，平面图就变得完全了。在此，我们列出了面完全图的完整目录：它们可分为七种类型。一个结果是，如果 q 是面完全平面图 G 中最大面的大小，那么 G 至少有 Floor[3/2 q] 个顶点。这个约束是已知的，但我们的证明与陈，格里尼和帕帕季米特留 1998 年的方法完全不同。我们的方法还得出了具有 n 个顶点的 2 连接面完整图的数量。我们还证明了，如果一个平面图最多有两个大小为 4 的面，而没有更大的面，那么在每个 4 面上加上两条对角线，就能得到一个可 5 色的图。
{"title":"A Catalog of Facially Complete Graphs","authors":"James Tilley, Stan Wagon, Eric Weisstein","doi":"arxiv-2409.11249","DOIUrl":"https://doi.org/arxiv-2409.11249","url":null,"abstract":"Considering regions in a map to be adjacent when they have nonempty\u0000intersection (as opposed to the traditional view requiring intersection in a\u0000linear segment) leads to the concept of a facially complete graph: a plane\u0000graph that becomes complete when edges are added between every two vertices\u0000that lie on a face. Here we present a complete catalog of facially complete\u0000graphs: they fall into seven types. A consequence is that if q is the size of\u0000the largest face in a plane graph G that is facially complete, then G has at\u0000most Floor[3/2 q] vertices. This bound was known, but our proof is completely\u0000different from the 1998 approach of Chen, Grigni, and Papadimitriou. Our method\u0000also yields a count of the 2-connected facially complete graphs with n\u0000vertices. We also show that if a plane graph has at most two faces of size 4\u0000and no larger face, then the addition of both diagonals to each 4-face leads to\u0000a graph that is 5-colorable.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A graph is called $K$-almost regular if its maximum degree is at most $K$ times the minimum degree. ErdH{o}s and Simonovits showed that for a constant $0< varepsilon< 1$ and a sufficiently large integer $n$, any $n$-vertex graph with more than $n^{1+varepsilon}$ edges has a $K$-almost regular subgraph with $n'geq n^{varepsilonfrac{1-varepsilon}{1+varepsilon}}$ vertices and at least $frac{2}{5}n'^{1+varepsilon}$ edges. An interesting and natural problem is whether there exits the spectral counterpart to ErdH{o}s and Simonovits's result. In this paper, we will completely settle this issue. More precisely, we verify that for constants $frac{1}{2}0$, if the spectral radius of an $n$-vertex graph $G$ is at least $cn^{varepsilon}$, then $G$ has a $K$-almost regular subgraph of order $n'geq n^{frac{2varepsilon^2-varepsilon}{24}}$ with at least $ c'n'^{1+varepsilon}$ edges, where $c'$ and $K$ are constants depending on $c$ and $varepsilon$. Moreover, for $0 frac{1}{2}$, $ex(n,mathcal{H}) = O(n^{1+xi})$ if and only if $spex(n,mathcal{H}) = O(n^xi)$.
{"title":"Almost regular subgraphs under spectral radius constrains","authors":"Weilun Xu, Guorong Gao, An Chang","doi":"arxiv-2409.10853","DOIUrl":"https://doi.org/arxiv-2409.10853","url":null,"abstract":"A graph is called $K$-almost regular if its maximum degree is at most $K$\u0000times the minimum degree. ErdH{o}s and Simonovits showed that for a constant\u0000$0< varepsilon< 1$ and a sufficiently large integer $n$, any $n$-vertex graph\u0000with more than $n^{1+varepsilon}$ edges has a $K$-almost regular subgraph with\u0000$n'geq n^{varepsilonfrac{1-varepsilon}{1+varepsilon}}$ vertices and at\u0000least $frac{2}{5}n'^{1+varepsilon}$ edges. An interesting and natural problem\u0000is whether there exits the spectral counterpart to ErdH{o}s and Simonovits's\u0000result. In this paper, we will completely settle this issue. More precisely, we\u0000verify that for constants $frac{1}{2}<varepsilonleq 1$ and $c>0$, if the\u0000spectral radius of an $n$-vertex graph $G$ is at least $cn^{varepsilon}$, then\u0000$G$ has a $K$-almost regular subgraph of order $n'geq\u0000n^{frac{2varepsilon^2-varepsilon}{24}}$ with at least $\u0000c'n'^{1+varepsilon}$ edges, where $c'$ and $K$ are constants depending on $c$\u0000and $varepsilon$. Moreover, for $0<varepsilonleqfrac{1}{2}$, there exist\u0000$n$-vertex graphs with spectral radius at least $cn^{varepsilon}$ that do not\u0000contain such an almost regular subgraph. Our result has a wide range of\u0000applications in spectral Tur'{a}n-type problems. Specifically, let\u0000$ex(n,mathcal{H})$ and $spex(n,mathcal{H})$ denote, respectively, the maximum\u0000number of edges and the maximum spectral radius among all $n$-vertex\u0000$mathcal{H}$-free graphs. We show that for $1geqxi > frac{1}{2}$,\u0000$ex(n,mathcal{H}) = O(n^{1+xi})$ if and only if $spex(n,mathcal{H}) =\u0000O(n^xi)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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