{"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}
{"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}
{"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}
{"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}
{"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
{"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}
{"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}
{"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}
Nina Chiarelli, Vesna Iršič, Marko Jakovac, William B. Kinnersley, Mirjana Mikalački
{"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}
{"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}
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