In this paper we further our understanding of the structure of class two cubic graphs, or snarks, as they are commonly known. We do this by investigating their 3-critical subgraphs, or as we will call them, minimal conflicting subgraphs. We consider how the minimal conflicting subgraphs of a snark relate to its possible minimal 4-edge-colourings. We fully characterise the relationship between the resistance of a snark and the set of minimal conflicting subgraphs. That is, we show that the resistance of a snark is equal to the minimum number of edges which can be selected from the snark, such that the selection contains at least one edge from each minimal conflicting subgraph. We similarly characterise the relationship between what we call \textit{the critical subgraph} of a snark and the set of minimal conflicting subgraphs. The critical subgraph being the set of all edges which are conflicting in some minimal colouring of the snark. Further to this, we define groups, or \textit{clusters}, of minimal conflicting subgraphs. We then highlight some interesting properties and problems relating to clusters of minimal conflicting subgraphs.
Over the past twenty years, critical and bicritical snarks have been appearing in the literature in various forms and in different contexts. Two main variants of criticality of snarks have been studied: criticality with respect to the non-existence of a $3$-edge-colouring and criticality with respect to the non-existence of a nowhere-zero $4$-flow. In this paper we show that these two kinds of criticality coincide, thereby completing previous partial results of de Freitas et al. [Electron. Notes Discrete Math. 50 (2015), 199--204] and Fiol et al. [ arXiv:1702.07156v1 (2017)].
In this paper we show that many snarks have shortest cycle covers of length $\frac{4}{3}m+c$ for a constant $c$, where $m$ is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length $\frac{4}{3}m+o(m)$. In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length $\frac{4}{3}m$ and satisfy the $(1,2)$-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to $\frac{4}{3}m$. We also prove some results for graphs with low oddness and discuss th…
The aim of this paper is to classify all snarks up to order $36$ and explain the reasons of their uncolourability. The crucial part of our approach is a computer-assisted structural analysis of cyclically $5$-connected critical snarks, which is justified by the fact that every other snark can be constructed from them by a series of simple operations while preserving uncolourability. Our results reveal that most of the analysed snarks are built up from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability. This sh…
In an orientation $O$ of the graph $G$, the edge $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, Hörsch and Szigeti defined the Frank number as the minimum $k$ for which $G$ admits $k$ orientations such that every edge $e$ of $G$ is deletable in at least one of the $k$ orientations. They conjectured the Frank number is at most $3$ for every $3$-edge-connected graph $G$. They proved the Petersen graph has Frank number $3$, but this was the only example with this property. We show an infinite class of graphs having Frank number $3$. Hörsch and Szi…
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
A note on vertex-critical induced subgraphs of shift graphs
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
For $k\geq 1$ and $n\geq 2k$, the Kneser graph $KG(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of $KG(n,k)$ is $n-2k+2$. Schrijver constructed a vertex-critical subgraph $SG(n,k)$ of $KG(n,k)$ with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for $k=2$ and arbitrary $n\geq 4$ by means of a nice explicit combi…
Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all $k\geq 4$, such graph contains $Ω(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles. In this paper, we mainly focus on 4-critical graphs and develop some novel tools for counting cy…
The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, $Γ_{\leq k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for $Γ_{\leq 2}$, and we use this forbidden subgraphs to get a classification of the graphs in $Γ_{\leq 2}$. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this ar…
An independent set $I_c$ is a \textit{critical independent set} if $|I_c| - |N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical independence number} of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the s…
An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The classes [h,2,1] are closed by taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, eve…
Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| \geq (\frac{k}{2} - \frac{1}{k-1})|V(G)| - \frac{k(k-3)}{2(k-1)}$. The class of graphs for which this bound is tight, $k$-Ore graphs, contain a notably large number of $K_{k-2}$-subgraphs. Subsequent work attempted to determine the asymptotic density for $k$-critical graphs that do \emph{not} contain large cliques as subgraphs, but only partial progress has been made on this problem. The second author showed that if $G$ is 5-cr…
We give an elementary proof for the non-$3$-colorability of $4$-chromatic Schrijver graphs thus providing such a proof also for $4$-chromatic Kneser graphs. To this end we use a complete description of the structure of $4$-chromatic Schrijver graphs that was already given by Braun and even earlier in an unpublished manuscript by Li. We also address connections to surface quadrangulations. In particular, we show that a spanning subgraph of $4$-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special …
For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $r$, such that any red/blue coloring of the edges of the graph $K_r$ contains either a red subgraph that is isomorphic to $G$ or a blue subgraph that is isomorphic to $H$. Let $S_k=K_{1,k}$ be a star of order $k+1$ and $K_n\sqcup S_k$ be a graph obtained from $K_n$ by adding a new vertex $v$ and joining $v$ to $k$ vertices of $K_n$. The star-critical Ramsey number $r_*(G,H)$ is the smallest positive integer $k$ such that any red/blue coloring of the edges of graph $K_{r-1}\sqcup S_k$ contains either a red s…
Fix $k \ge 2$ and let $H$ be a graph with $χ(H) = k+1$ containing a critical edge. We show that for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian.
Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one of the middle vertices of the $P_4$. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. In this paper, we prove that there are finitely many 5-vertex-critical $(P_5,chair)$-free graphs.
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number $χ_t(G)$ is the least integer $k$ for which $G$ admits a coloring with $k$ colors such that each color class induces a $(t-1)$-degenerate subgraph of $G$. So $χ_1$ is the chromatic number and $χ_2$ is the point aboricity. The point partition number $χ_t$ with $t\geq 1$ was introduced by Lick and White. A graph $G$ is called $χ_t$-critical if every proper subgraph $H$ of $G$ satisfies $χ_t(H)<χ_t(G)$. In this paper we prove that if $G$ is a $χ_t$-critical graph whose orde…
A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a graph $G$ is \emph{$k$-list-critical} if $G$ is not $(k-1)$-choosable, but every proper subgraph of $G$ is $(k-1)$-choosable. The problem of bounding (from below) the number of edges in a $k$-critical graph has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem.…
Let $G$ be a simple graph. Denote by $n$, $Δ(G)$ and $χ' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > Δ(G)$, and {\it critical} if $χ'(H) < χ'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $χ'(G) = Δ(G)+1$. The \emph{core} of $G$, denoted by $G_Δ$, is the subgraph of $G$ induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attacking the overfull conjecture. Along this direction, we in thi…
Let $G$ be a simple graph, and let $n$, $Δ(G)$ and $χ' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > Δ(G)$, and critical if $χ'(H) < χ'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $χ'(G) = Δ(G)+1$. The core of $G$, denoted by $G_Δ$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $Δ(G) \ge 4$, if the maximum degree of $G_Δ$ is at most two, then $G$ is overfull, which in turn gives $Δ…
Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k \ge Δ+1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $Δ$ denotes the maximum degree of $G$. Tashkinov tree method shows that critical graphs contain a subgraph with two important properties named closed and elementary. Recently, efforts have been made in extending graphs beyond Tashkinov trees. However, these results can only keep one of the two essential properties. I…
Given a graph $G$, denote by $Δ$ and $χ^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian.
Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_t$ be the path and the cycle on $t$ vertices, respectively. A bull is the graph obtained from a triangle with two disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical ($P_5$,bull)-free graphs.
Three-coloring triangle-free graphs without long forbidden paths
College of Computer Science, Nankai University（南开大学计算机学院）；Department of Computer Science, KU Leuven Campus Kulak-Kortrijk（荷鲁文大学库勒-科特赖克校区计算机科学系）；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学和统计系）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
Counting color-critical subgraphs under Nikiforov's condition
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）；School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）
A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)log n/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.
The {\em chromatic edge-stability number} $es_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em $(3,2)$-critical} if $χ(G)=3$, $es_χ(G)=2$ and for any edge $e\in E(G)$, $es_χ(G-e)<es_χ(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].
Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, denote by $\mathcal{MG}_t$ the class of graphs whose maximum multiplicity is at most $t$. A graph $G$ is called strictly $t$-degenerate if every non-empty subgraph $H$ of $G$ contains a vertex $v$ whose degree in $H$ is at most $t-1$. The point partition number $χ_t(G)$ of $G$ is smallest number of colors needed to color the vertices of $G$ so that each vertex receives a color and vertices with the same color induce a strictly $t$-degenerate subgraph of $G$. So $χ_1$ is …
We prove that every $k$-list-critical graph ($k \ge 7$) on $n \ge k+2$ vertices has at least $\frac12 \left(k-1 + \frac{k-3}{(k-c)(k-1) + k-3}\right)n$ edges where $c = (k-3)\left(\frac12 - \frac{1}{(k-1)(k-2)}\right)$. This improves the bound established by Kostochka and Stiebitz. The same bound holds for online $k$-list-critical graphs, improving the bound established by Riasat and Schauz. Both bounds follow from a more general result stating that either a graph has many edges or it has an Alon-Tarsi orientable induced subgraph satisfying a certain degree condition.
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $ϕ$ of $J$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $J$. We prove a conjecture of Dvorak et al. that if $H$ is a minimal subgraph of $G$ such that $C$ is a subgraph of $H$ and every $L$-coloring of $C$ that extends to an $L$-coloring of $H$ also extends to an $L$-coloring of $G$, then $|V(H)|\le 19|V(C)|$. This is a lemma that plays an important role in subsequent papers, becau…
The chromatic edge-stability number ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $χ(G')=χ(G)-1$. Edge-stability critical graphs are introduced as the graphs $G$ with the property that ${\rm es}_χ(G-e) < {\rm es}_χ(G)$ holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph with $χ(G)=k$ and ${\rm es}_χ(G)=\ell$, then $G$ is $(k,\ell)$-critical. Graphs which are $(3,2)$-critical and contain at most four odd cycles are classified. It is also proved that the problem of deciding whether a graph $G$ has $χ(…
The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For each of these four related parameters, we present some simple characterizations of the graphs attaining the maximum possible values. As a main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-orde…
A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex $k$-critical graph. This is widely open for any integer $k\geq 4$. Using a structural characterization of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz proved in 1987 that for $k\geq 4$ and sufficiently large $n$, this maximum number is less than the number of edges in the $n$-vertex balanced complete $(k-2)$-partite graph. In this paper we obtai…
Given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph in color $i$ isomorphic to $H_{i}$ for some $i\in\{1,2,\ldots,c\}$. The critical graphs for $r(H_1,H_2,\ldots,H_c)$ are edge-colored complete graphs on $r(H_1,H_2,\ldots,H_c)-1$ vertices with $c$ colors which contain no subgraphs in color $i$ isomorphic to $H_{i}$ for any $i\in \{1,2,\ldots,c\}$. For $n_1\geq n_2\geq \ldots\geq n_c\geq 1$, Cockayne and Lorimer (The Ramsey number for stripes,…
This work introduces the concept of \emph{upper-critical graphs}, in a complementary way of the conventional (lower)critical graphs: an element $x$ of a graph $G$ is called \emph{critical} if $χ(G-x)<χ(G)$. It is said that $G$ is a \emph{critical graph} if every element (vertex or edge) of $G$ is critical. Analogously, a graph $G$ is called \emph{upper-critical} if there is no edge that can be added to $G$ such that $G$ preserves its chromatic number, i.e. \{$e \in E(\bar{G}) \; | \; χ(G+e) = χ(G)$ \} $=$ $\emptyset$. We show that the class of upper-critical graphs is the same as the class of …
Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|G_n|, with probability going to one, uniformly in p. The method from Ajtai, Komlos and Szemeredi [Combinatorica 2 (1982) 1-7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of…
A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to $H$-free graphs, that is, graphs with no induced subgraph isomorphic to $H$. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1.
A $c$-crossing-critical graph is one that has crossing number at least $c$ but each of its proper subgraphs has crossing number less than $c$. Recently, a set of explicit construction rules was identified by Bokal, Oporowski, Richter, and Salazar to generate all large $2$-crossing-critical graphs (i.e., all apart from a finite set of small sporadic graphs). They share the property of containing a generalized Wagner graph $V_{10}$ as a subdivision. In this paper, we study these graphs and establish their order, simple crossing number, edge cover number, clique number, maximum degree, chromatic …
