For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$ denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains an $F$-free induced subgraph on $m$ vertices. This function has been extensively studied in the last 60 years when $F$ and $H$ are cliques and became known as the ErdH{o}s-Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstra"ete initiated the systematic study of this function in the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstra"ete, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $varepsilon_F>0$ such that $f_{F,K_r}(n)=O(n^{1/2-varepsilon_F})$. This result is tight in two ways. Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph. Secondly, we show that for all $rgeq 4$ and $varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Omega(n^{1/2-varepsilon})$. Along the way of proving this, we show in particular that for every graph $F$ with minimum degree $t$, we have $f_{F,K_4}(n)=Omega(n^{1/2-6/sqrt{t}})$. This answers (in a strong form) another question of Mubayi and Verstra"ete. Finally, we prove that there exist absolute constants $0
{"title":"Induced subgraphs of $K_r$-free graphs and the Erdős--Rogers problem","authors":"Lior Gishboliner, Oliver Janzer, Benny Sudakov","doi":"arxiv-2409.06650","DOIUrl":"https://doi.org/arxiv-2409.06650","url":null,"abstract":"For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$\u0000denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains\u0000an $F$-free induced subgraph on $m$ vertices. This function has been\u0000extensively studied in the last 60 years when $F$ and $H$ are cliques and\u0000became known as the ErdH{o}s-Rogers function. Recently, Balogh, Chen and Luo,\u0000and Mubayi and Verstra\"ete initiated the systematic study of this function in\u0000the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstra\"ete, we prove\u0000that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there\u0000exists some $varepsilon_F>0$ such that\u0000$f_{F,K_r}(n)=O(n^{1/2-varepsilon_F})$. This result is tight in two ways.\u0000Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph.\u0000Secondly, we show that for all $rgeq 4$ and $varepsilon>0$, there exists a\u0000$K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Omega(n^{1/2-varepsilon})$.\u0000Along the way of proving this, we show in particular that for every graph $F$\u0000with minimum degree $t$, we have $f_{F,K_4}(n)=Omega(n^{1/2-6/sqrt{t}})$.\u0000This answers (in a strong form) another question of Mubayi and Verstra\"ete.\u0000Finally, we prove that there exist absolute constants $0<c<C$ such that for\u0000each $rgeq 4$, if $F$ is a bipartite graph with sufficiently large minimum\u0000degree, then $Omega(n^{frac{c}{log r}})leq f_{F,K_r}(n)leq\u0000O(n^{frac{C}{log r}})$. This shows that for graphs $F$ with large minimum\u0000degree, the behaviour of $f_{F,K_r}(n)$ is drastically different from that of\u0000the corresponding off-diagonal Ramsey number $f_{K_2,K_r}(n)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197627","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}
This paper presents a possible link between Cages and Expander Graphs by introducing three interconnected variants of the Bermond and Bollob'as Conjecture, originally formulated in 1981 within the context of the Degree/Diameter Problem. We adapt these conjectures to cages, with the most robust variant posed as follows: Does there exist a constant $c$ such that for every pair of parameters $(k,g)$ there exists a $k$-regular graph of girth $g$ and order not exceeding $ M(k,g) + c $?; where $M(k,g)$ denotes the value of the so-called Moore bound for cages. We show that a positive answer to any of the three variants of the Bermond and Bollob'as Conjecture for cages considered in our paper would yield expander graphs (expander families); thereby establishing a connection between Cages and Expander Graphs.
{"title":"Analogues of Bermond-Bollobás Conjecture for Cages Yield Expander Families","authors":"Leonard Chidiebere Eze, Robert Jajcay","doi":"arxiv-2409.06629","DOIUrl":"https://doi.org/arxiv-2409.06629","url":null,"abstract":"This paper presents a possible link between Cages and Expander Graphs by\u0000introducing three interconnected variants of the Bermond and Bollob'as\u0000Conjecture, originally formulated in 1981 within the context of the\u0000Degree/Diameter Problem. We adapt these conjectures to cages, with the most\u0000robust variant posed as follows: Does there exist a constant $c$ such that for\u0000every pair of parameters $(k,g)$ there exists a $k$-regular graph of girth $g$\u0000and order not exceeding $ M(k,g) + c $?; where $M(k,g)$ denotes the value of\u0000the so-called Moore bound for cages. We show that a positive answer to any of\u0000the three variants of the Bermond and Bollob'as Conjecture for cages\u0000considered in our paper would yield expander graphs (expander families);\u0000thereby establishing a connection between Cages and Expander Graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225151","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}
Paul Bastide, Clément Legrand-Duchesne, Alp Müyesser
A seminal result of Koml'os, S'ark"ozy, and Szemer'edi states that any n-vertex graph G with minimum degree at least (1/2 + {alpha})n contains every n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended this result to show that such graphs G in fact support an optimally spread distribution on copies of a given T, which implies, using the recent breakthroughs on the Kahn-Kalai conjecture, the robustness result that T is a subgraph of sparse random subgraphs of G as well. Pham, Sah, Sawhney, and Simkin construct their optimally spread distribution by following closely the original proof of the Koml'os-S'ark"ozy-Szemer'edi theorem which uses the blow-up lemma and the Szemer'edi regularity lemma. We give an alternative, regularity-free construction that instead uses the Koml'os-S'ark"ozy-Szemer'edi theorem (which has a regularity-free proof due to Kathapurkar and Montgomery) as a black-box. Our proof is based on the simple and general insight that, if G has linear minimum degree, almost all constant sized subgraphs of G inherit the same minimum degree condition that G has.
Koml'os, S'ark"ozy, and Szemer'edi 的一个开创性结果指出，任何最小度至少为 (1/2 + {alpha})n 的 n 顶点图 G 都包含每一棵有界度的 n 顶点树 T。最近，Pham、Sah、Sawhney 和 Simkine 扩展了这一结果，证明此类图 G 事实上支持给定 T 的副本的最优分布，这意味着，利用最近对 Kahn-Kalai 猜想的突破，T 也是 G 的稀疏随机子图的子图的鲁棒性结果。Pham、Sah、Sawhney 和 Simkin 紧跟 Koml'os-S'ark"ozy-Szemer'edi 定理的原始证明，利用炸毁lemma 和 Szemer'edi regularity lemma，构建了他们的最优分布。我们给出了另一种无正则性的构造，它使用Koml'os-S'ark"ozy-Szemer'edi theorem（卡塔普卡尔和蒙哥马利对它进行了无正则性证明）作为黑箱。我们的证明基于一个简单而普遍的见解，即如果 G 具有线性最小度，那么 G 的几乎所有常量子图都继承了与 G 相同的最小度条件。
{"title":"Random embeddings of bounded degree trees with optimal spread","authors":"Paul Bastide, Clément Legrand-Duchesne, Alp Müyesser","doi":"arxiv-2409.06640","DOIUrl":"https://doi.org/arxiv-2409.06640","url":null,"abstract":"A seminal result of Koml'os, S'ark\"ozy, and Szemer'edi states that any\u0000n-vertex graph G with minimum degree at least (1/2 + {alpha})n contains every\u0000n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin\u0000extended this result to show that such graphs G in fact support an optimally\u0000spread distribution on copies of a given T, which implies, using the recent\u0000breakthroughs on the Kahn-Kalai conjecture, the robustness result that T is a\u0000subgraph of sparse random subgraphs of G as well. Pham, Sah, Sawhney, and\u0000Simkin construct their optimally spread distribution by following closely the\u0000original proof of the Koml'os-S'ark\"ozy-Szemer'edi theorem which uses the\u0000blow-up lemma and the Szemer'edi regularity lemma. We give an alternative,\u0000regularity-free construction that instead uses the\u0000Koml'os-S'ark\"ozy-Szemer'edi theorem (which has a regularity-free proof due\u0000to Kathapurkar and Montgomery) as a black-box. Our proof is based on the simple\u0000and general insight that, if G has linear minimum degree, almost all constant\u0000sized subgraphs of G inherit the same minimum degree condition that G has.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197628","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 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration possesses a remarkable structure. More precisely, it can be expressed as a linear combination of exponentials $m^{2g-2+2d}$, where $m$ ranges over the integers from $1$ to $binom{d}{2}$. In this paper, we generalise this structural result to Hurwitz numbers that enumerate branched covers which also have a prescribed ramification profile over one point. Our proof fundamentally uses the infinite wedge space, in particular the connected correlators of products of $mathcal{E}$-operators. The recent study of Hurwitz numbers has often focussed on their structure with fixed genus and varying ramification profile. Our main result is orthogonal to this, allowing for the explicit calculation and the asymptotic analysis of Hurwitz numbers in large genus. We pose the broad question of which other enumerative problems exhibit analogous structure. We prove that orbifold Hurwitz numbers can also be expressed as a linear combination of exponentials and conjecture that monotone Hurwitz numbers share a similar structure, but with the inclusion of an additional linear term.
{"title":"The structure of Hurwitz numbers with fixed ramification profile and varying genus","authors":"Norman Do, Jian He, Heath Robertson","doi":"arxiv-2409.06655","DOIUrl":"https://doi.org/arxiv-2409.06655","url":null,"abstract":"In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$,\u0000branched covers of the Riemann sphere with simple ramification over prescribed\u0000points and no branching elsewhere. He showed that for fixed degree $d$, the\u0000enumeration possesses a remarkable structure. More precisely, it can be\u0000expressed as a linear combination of exponentials $m^{2g-2+2d}$, where $m$\u0000ranges over the integers from $1$ to $binom{d}{2}$. In this paper, we generalise this structural result to Hurwitz numbers that\u0000enumerate branched covers which also have a prescribed ramification profile\u0000over one point. Our proof fundamentally uses the infinite wedge space, in\u0000particular the connected correlators of products of $mathcal{E}$-operators.\u0000The recent study of Hurwitz numbers has often focussed on their structure with\u0000fixed genus and varying ramification profile. Our main result is orthogonal to\u0000this, allowing for the explicit calculation and the asymptotic analysis of\u0000Hurwitz numbers in large genus. We pose the broad question of which other enumerative problems exhibit\u0000analogous structure. We prove that orbifold Hurwitz numbers can also be\u0000expressed as a linear combination of exponentials and conjecture that monotone\u0000Hurwitz numbers share a similar structure, but with the inclusion of an\u0000additional linear term.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225150","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 the last forty years, many scientists used graph theory to develop mathematical models for analyzing structures and properties of various chemical compounds. In this paper, we will establish formulas and bounds for generalized first Zagreb Index and coindex, which are based on degrees of vertices. In addition, for triangle and quadrangle free graphs, we will establish formulas and bounds for generalized first leap Zagreb Index and coindex, which are based on 2-distance degrees of vertices. Additionally, we will establish sharp bounds of generalized first Zagreb index and the leap index for various types of graphs and provide examples for which the sharp bounds are attained. In addition, we will find regression models and compare the first Zagreb index and the first leap Zagreb index for predicting some physicochemical properties of certain chemical compounds, benzenoid hydrocarbons.
{"title":"Sharp Bounds for Generalized Zagreb Indices of Graphs","authors":"Sanju Vaidya, Jeff Chang","doi":"arxiv-2409.06081","DOIUrl":"https://doi.org/arxiv-2409.06081","url":null,"abstract":"In the last forty years, many scientists used graph theory to develop\u0000mathematical models for analyzing structures and properties of various chemical\u0000compounds. In this paper, we will establish formulas and bounds for generalized\u0000first Zagreb Index and coindex, which are based on degrees of vertices. In\u0000addition, for triangle and quadrangle free graphs, we will establish formulas\u0000and bounds for generalized first leap Zagreb Index and coindex, which are based\u0000on 2-distance degrees of vertices. Additionally, we will establish sharp bounds\u0000of generalized first Zagreb index and the leap index for various types of\u0000graphs and provide examples for which the sharp bounds are attained. In\u0000addition, we will find regression models and compare the first Zagreb index and\u0000the first leap Zagreb index for predicting some physicochemical properties of\u0000certain chemical compounds, benzenoid hydrocarbons.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197636","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}
Let $d,k$ be natural numbers and let $mathcal{L}_1, dots, mathcal{L}_k in mathrm{GL}_d(mathbb{Q})$ be linear transformations such that there are no non-trivial subspaces $U, V subseteq mathbb{Q}^d$ of the same dimension satisfying $mathcal{L}_i(U) subseteq V$ for every $1 leq i leq k$. For every non-empty, finite set $A subset mathbb{R}^d$, we prove that [ |mathcal{L}_1(A) + dots + mathcal{L}_k(A) | geq k^d |A| - O_{d,k}(|A|^{1- delta}), ] where $delta >0$ is some absolute constant depending on $d,k$. Building on work of Conlon-Lim, we can show stronger lower bounds when $k$ is even and $mathcal{L}_1, dots, mathcal{L}_k$ satisfy some further incongruence conditions, consequently resolving various cases of a conjecture of Bukh. Moreover, given any $d, kin mathbb{N}$ and any finite, non-empty set $A subset mathbb{R}^d$ not contained in a translate of some hyperplane, we prove sharp lower bounds for the cardinality of the $k$-fold sumset $kA$ in terms of $d,k$ and $|A|$. This can be seen as a $k$-fold generalisation of Freiman's lemma.
{"title":"Brunn-Minkowski type estimates for certain discrete sumsets","authors":"Albert Lopez Bruch, Yifan Jing, Akshat Mudgal","doi":"arxiv-2409.05638","DOIUrl":"https://doi.org/arxiv-2409.05638","url":null,"abstract":"Let $d,k$ be natural numbers and let $mathcal{L}_1, dots, mathcal{L}_k in\u0000mathrm{GL}_d(mathbb{Q})$ be linear transformations such that there are no\u0000non-trivial subspaces $U, V subseteq mathbb{Q}^d$ of the same dimension\u0000satisfying $mathcal{L}_i(U) subseteq V$ for every $1 leq i leq k$. For\u0000every non-empty, finite set $A subset mathbb{R}^d$, we prove that [\u0000|mathcal{L}_1(A) + dots + mathcal{L}_k(A) | geq k^d |A| - O_{d,k}(|A|^{1-\u0000delta}), ] where $delta >0$ is some absolute constant depending on $d,k$.\u0000Building on work of Conlon-Lim, we can show stronger lower bounds when $k$ is\u0000even and $mathcal{L}_1, dots, mathcal{L}_k$ satisfy some further\u0000incongruence conditions, consequently resolving various cases of a conjecture\u0000of Bukh. Moreover, given any $d, kin mathbb{N}$ and any finite, non-empty set\u0000$A subset mathbb{R}^d$ not contained in a translate of some hyperplane, we\u0000prove sharp lower bounds for the cardinality of the $k$-fold sumset $kA$ in\u0000terms of $d,k$ and $|A|$. This can be seen as a $k$-fold generalisation of\u0000Freiman's lemma.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197640","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}
Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu
In this note, we show that extremal Szemer'{e}di-Trotter configurations are rigid in the following sense: If $P,L$ are sets of points and lines determining at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$ of points of size at most $k = k_0(C)$ such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth-Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester-Gallai configurations.
{"title":"On the structure of extremal point-line arrangements","authors":"Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu","doi":"arxiv-2409.06115","DOIUrl":"https://doi.org/arxiv-2409.06115","url":null,"abstract":"In this note, we show that extremal Szemer'{e}di-Trotter configurations are\u0000rigid in the following sense: If $P,L$ are sets of points and lines determining\u0000at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$\u0000of points of size at most $k = k_0(C)$ such that, heuristically, fixing those\u0000points fixes a positive fraction of the arrangement. That is, the incidence\u0000structure and a small number of points determine a large part of the\u0000arrangement. The key tools we use are the Guth-Katz polynomial partitioning,\u0000and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show\u0000the rigidity of near-Sylvester-Gallai configurations.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197635","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 show that if $mathcal{A} subset {[n] choose n/2}$ with measure bounded away from zero and from one, then the $Omega(sqrt{n})$-iterated upper shadows of $mathcal{A}$ and $mathcal{A}^c$ intersect in a set of positive measure. This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a stability result for the Kruskal--Katona theorem.
{"title":"Intersections of iterated shadows","authors":"Hou Tin Chau, David Ellis, Marius Tiba","doi":"arxiv-2409.05487","DOIUrl":"https://doi.org/arxiv-2409.05487","url":null,"abstract":"We show that if $mathcal{A} subset {[n] choose n/2}$ with measure bounded\u0000away from zero and from one, then the $Omega(sqrt{n})$-iterated upper shadows\u0000of $mathcal{A}$ and $mathcal{A}^c$ intersect in a set of positive measure.\u0000This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a\u0000stability result for the Kruskal--Katona theorem.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197641","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 said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. ErdH{o}s, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.
{"title":"Infinitely many minimally Ramsey size-linear graphs","authors":"Yuval Wigderson","doi":"arxiv-2409.05931","DOIUrl":"https://doi.org/arxiv-2409.05931","url":null,"abstract":"A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for\u0000every graph $H$ with no isolated vertices. ErdH{o}s, Faudree, Rousseau, and\u0000Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper\u0000subgraphs is, and they asked whether there exist infinitely many such graphs.\u0000In this short note, we answer this question in the affirmative.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197637","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}
An textit{$(n,m)$-graph} $G$ is a graph having both arcs and edges, and its arcs (resp., edges) are labeled using one of the $n$ (resp., $m$) different symbols. An textit{$(n,m)$-complete graph} $G$ is an $(n,m)$-graph without loops or multiple edges in its underlying graph such that identifying any pair of vertices results in a loop or parallel adjacencies with distinct labels. We show that a planar $(n,m)$-complete graph cannot have more than $3(2n+m)^2+(2n+m)+1$ vertices, for all $(n,m) neq (0,1)$ and the bound is tight. This answers a naturally fundamental extremal question in the domain of homomorphisms of $(n,m)$-graphs and positively settles a recent conjecture by Bensmail textit{et al.}~[Graphs and Combinatorics 2017]. Essentially, our result finds the clique number for planar $(n,m)$-graphs, which is a difficult problem except when $(n,m)=(0,1)$, answering a sub-question to finding the chromatic number for the family of planar $(n,m)$-graphs.
一个（textit{$(n,m)$图} $G$是一个既有弧又有边的图，其弧（或边）用$n$（或$m$）个不同符号中的一个来标注。一个文本{$(n,m)$完整图}$G$是一个$(n,m)$图，在它的底层图中没有循环或多条边，这样识别任何一组顶点都会导致循环或带有不同标签的平行邻接。我们发现，对于所有 $(n,m) neq (0,1)$，一个平面$(n,m)$完整图不可能有超过$3(2n+m)^2+(2n+m)+1$的顶点，而且这个约束是严格的。这回答了$(n,m)$图的同态领域中一个天然的基本极值问题，并正面解决了本斯梅尔（Bensmail textit{et al.}~[Graphs and Combinatorics 2017]最近提出的一个猜想。从本质上讲，我们的结果找到了平面$(n,m)$图的簇数（这是一个难题，除非当$(n,m)=(0,1)$时），回答了找到平面$(n,m)$图族的色数的子问题。
{"title":"A step towards finding the analog of the Four-Color Theorem for $(n,m)$-graphs","authors":"Susobhan Bandopadhyay, Sagnik Sen, S Taruni","doi":"arxiv-2409.05678","DOIUrl":"https://doi.org/arxiv-2409.05678","url":null,"abstract":"An textit{$(n,m)$-graph} $G$ is a graph having both arcs and edges, and its\u0000arcs (resp., edges) are labeled using one of the $n$ (resp., $m$) different\u0000symbols. An textit{$(n,m)$-complete graph} $G$ is an $(n,m)$-graph without\u0000loops or multiple edges in its underlying graph such that identifying any pair\u0000of vertices results in a loop or parallel adjacencies with distinct labels. We\u0000show that a planar $(n,m)$-complete graph cannot have more than\u0000$3(2n+m)^2+(2n+m)+1$ vertices, for all $(n,m) neq (0,1)$ and the bound is\u0000tight. This answers a naturally fundamental extremal question in the domain of\u0000homomorphisms of $(n,m)$-graphs and positively settles a recent conjecture by\u0000Bensmail textit{et al.}~[Graphs and Combinatorics 2017]. Essentially, our\u0000result finds the clique number for planar $(n,m)$-graphs, which is a difficult\u0000problem except when $(n,m)=(0,1)$, answering a sub-question to finding the\u0000chromatic number for the family of planar $(n,m)$-graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225194","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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