Let $G$ be a simple graph with maximum degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. We say $G$ is of \emph{Class 1} if $χ'(G)=Δ$, and is of \emph{Class 2} otherwise. A graph $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ+1$ and $χ'(H)<Δ+1$ for every proper subgraph $H$ of $G$, and is \emph{overfull} if $|E(G)|>Δ\lfloor (|V(G)|-1)/2 \rfloor$. Clearly, overfull graphs are Class 2. Hilton and Zhao in 1997 conjectured that if $G$ is obtained from an $n$-vertex $Δ$-regular Class 1 graph with maximum degree greater than $n/3$ by splitting a vertex, then being overfull is the only reason for $G$ to be Class 2. This conjecture was only confirmed when $Δ\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. In this paper, we improve the bound on $Δ$ from $\frac{n}{2}(\sqrt{7}-1)$ to $0.75n$. Considering the structure of $Δ$-critical graphs with a vertex of degree 2, we also show that for an $n$-vertex $Δ$-critical graph with $Δ\ge \frac{3n}{4}$, if it contains a vertex of degree 2, then it is overfull. We actually obtain a more general form of this result, which partially supports the overfull conjecture of Chetwynd and Hilton from 1986, which states that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ>n/3$, then $G$ contains an overfull subgraph $H$ with $Δ(H)=Δ$. Our proof techniques are new and might shed some light on attacking both of the conjectures when $Δ$ is large.
We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every $3$-uniform $n$-vertex ($n$ even) hypergraph $\mathcal{H}$ with minimum vertex degree $δ_1(\mathcal{H})\geq \left(\frac7{16}+o(1)\right)\binom{n}{2}$ contains a loose Hamilton cycle. This bound is asymptotically best possible.
We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically optimal.
In 1997 Klavžar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.
In this paper, we consider a general degree sum condition sufficient to imply the existence of $k$ vertex-disjoint chorded cycles in a graph $G$. Let $σ_t(G)$ be the minimum degree sum of $t$ independent vertices of $G$. We prove that if $G$ is a graph of sufficiently large order and $σ_t(G)\geq 3kt-t+1$ with $k\geq 1$, then $G$ contains $k$ vertex-disjoint chorded cycles. We also show that the degree sum condition on $σ_t(G)$ is sharp. To do this, we also investigate graphs without chorded cycles.
We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.
We prove that the vertex degree threshold for tiling $\C_4^3$ (the 3-uniform hypergraph with four vertices and two triples) in a 3-uniform hypergraph on $n\in 4\mathbb N$ vertices is $\binom{n-1}2 - \binom{\frac34 n}2+\frac38n+c$, where $c=1$ if $n\in 8\mathbb N$ and $c=-\frac12$ otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.
We study theorems giving sufficient conditions on the vertex degrees of a graph $G$ to guarantee $G$ is $t$-tough. We first give a best monotone theorem when $t\ge1$, but then show that for any integer $k\ge1$, a best monotone theorem for $t=\frac1k\le 1$ requires at least $f(k)\cdot|V(G)|$ nonredundant conditions, where $f(k)$ grows superpolynomially as $k\rightarrow\infty$. When $t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms of its vertex degrees.
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $φ$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{φ(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the…
Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of $ϕ^2$, where $ϕ$ is the golden ratio. The exponent $ϕ^2$ is best possible. This inequality is generalized for all graphs with bounded maximum average degree.
Let $γ(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $γ(G-v)=γ(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this paper, we correspondingly introduce two such definitions: (i) a set $S\subseteq V(G)$ is called a \emph{strong critical vertex-set} of $G$ if $γ(G-S)=γ(G)-|S|$; (ii) a graph $G$ is called \emph{strong $l$-vertex-sets-critical} if $V(G)$ can be partitioned into $l$ strong critical vertex-sets of $G$. Whereafter, we give some properties of strong $l$-vertex-sets-critical graph…
For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $π=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invarian…
Continuing the recent work of L. Zhong and K. Xu [MATCH Commun. Math. Comput. Chem.71(2014) 627-642], we determine inequalities among several vertex-degree-based topological indices; first geometric-arithmetic index(GA), augmented Zagreb index (AZI), Randi$\acute{c}$ index (R), atom-bond connectivity index (ABC), sum-connectivity index (X)and harmonic index (H).
We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-graph without isolated vertex. More precisely, suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight.
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves.
We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buß, Hàn and Schacht who proved the corresponding asymptotical result.
Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least v(G)/(s+1), (a2) if G is a graph with every vertex of degree at least two and at most 3 and G has no 5-vertex components, then t(G,2) is at least v(G)/4, (a3) if G is a graph with every vertex of degree at least one and at most s and G has no k--vertex component, where k >1 and s > 2, then t(G,k) is at least (v(G) - k)/(sk - k +1), and (a4) the above bounds are…
In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result.
The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $χ(G-S)=χ(G)-1$. In this paper we construct infinitely many graphs $G$ with $Δ(G)=4$, $χ(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.
A graph $G$ is said to be $k$-$γ$-vertex critical if the domination numbers $γ(G)$ of $G$ is $k$ and $γ(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$γ_{c}$-vertex critical if the connected domination numbers $γ_{c}(G)$ of $G$ is $k$ and $γ_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is to determine whether or not $2$-connected $k$-$γ$-vertex critical graphs are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected $k$-$γ$-vertex critical graph which is non-Hamiltonian. We prove that every $2$-connected $3$-$γ$-vert…
The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of all $n$-vertex trees, $\mathbb{T}_{n,Δ}$ be the set of $n$-vertex trees with given maximum degree equal to $Δ$, $\mathbb{T}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree, and let $\mathbb{T}_{n,Δ}^k$ be the set of $n$-vertex trees with exactly $k$ vertices of given maximum degree equal to $Δ.$ In this paper, we first determine…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss some possible extension to maps of higher genus.
A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and Mel'nikov [Vertex oblique graphs, same proceedings] have constructed infinite classes of {\em super vertex-oblique} graphs, where the degree-types of $G$ are distinct even from the degree types of $\bar{G}$. $G$ is vertex oblique iff $\bar{G}$ is; but $G$ and $\bar{G}$ cannot be isomorphic, since self-complementary graphs always have non-trivial automorphism…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers $D$. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and…
Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.
Several games that arise from graph coloring have been introduced and studied. Let $φ$ denote a graph invariant that arises from such a game. If $G$ is a graph and $φ(G-x)\neq φ(G)=k$, $k \geq 1$, holds true for every vertex $x \in V(G)$, then $G$ is called a $k$-$φ$-game-vertex-critical graph. We study the concept of $φ$-game-vertex-criticality for $φ\in \{χ_g, χ_i, χ_{ig}^{A}, χ_{ig}^{AB}\}$, where $χ_g$ denotes the standard game chromatic number, $χ_i$ denotes the indicated game chromatic number and $χ_{ig}^{A}$, $χ_{ig}^{AB}$ denote two versions of the independence game chromatic number. S…
For a $k$-uniform hypergraph $H$, let $δ_1(H)$ denote the minimum vertex degree of $H$, and $ν(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $β>0$, there exists an integer $n_0(β,k)$ such that for positive integers $n\geq n_0$ and $m\leq (\frac{k}{2(k-1)}-β)\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with $δ_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $ν(H)\geq m$. This improves upon earlier results of Bollobás, Daykin and Erdős (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.
A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if …
Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular, for $d_+\in \left(\sqrt{dn},n-1\right]$, we show the existence of a vertex in $G$ of degree between $d_+-\left(\frac{(d_+-d)n}{n-d_++\sqrt{d_+^2-dn}}\right)$ and $d_+$.
A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by $γ(G)$. A graph $G$ is said to be $γ$- vertex-critical if $γ(G-v)< γ(G)$, for every vertex $v$ in $G$. Let $G$ be a 2-connected $K_{1,5}$-free 3-vertex-critical graph. For any vertex $v \in V(G)$, we show that $G-v$ has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer.
A classical result of Komlós, Sárközy and Szemerédi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on the degree conditions are sharp. On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every $n$-vertex graph $G_α$ with minimum degree at least $αn$ for any fixed $α>0$ and every $n$-vertex tree $T$ with bounded maximum degree, one can still find a copy of $T$ in $G_α$ with high probability after adding $O(n)$ randomly-chosen edges to $G_α$. We ex…
In 1963, Corrádi and Hajnal settled a conjecture of Erdős by proving that, for all $k \geq 1$, any graph $G$ with $|G| \geq 3k$ and minimum degree at least $2k$ contains $k$ vertex-disjoint cycles. In 2008, Finkel proved that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum degree at least $3k$ contains $k$ vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum Ore-degree at least $6k-1$ contains $k$ vertex-disjoint cycles. We r…
In 2006, Noga Alon raised the following open problem: Does there exist an absolute constant $c>0$ such that every $2n$-vertex digraph with minimum out-degree at least $s$ contains an $n$-vertex subdigraph with minimum out-degree at least $\frac{s}{2}-c$ ? In this note, we answer this natural question in the negative, by showing that for arbitrarily large values of $n$ there exists a $2n$-vertex tournament with minimum out-degree $s=n-1$, in which every $n$-vertex subdigraph contains a vertex of out-degree at most $\frac{s}{2}-\left(\frac{1}{2}+o(1)\right)\log_3(s)$.
The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is the skeleton of the matching polytope of G, denoted G(M(G)). Since the number of vertices of G(M(G)) is huge, the structural properties of these graphs have been studied in particular classes. In this paper, for an arbitrary graph G, we obtain a formulae to compute the degree of a vertex of G(M(G)) and prove that the minimum degree of G(M(G)) is equal to the number of edges of G. Also, we identify…
Given positive integers $a\leq b \leq c$, let $K_{a,b,c}$ be the complete 3-partite 3-uniform hypergraph with three parts of sizes $a,b,c$. Let $H$ be a 3-uniform hypergraph on $n$ vertices where $n$ is divisible by $a+b+c$. We asymptotically determine the minimum vertex degree of $H$ that guarantees a perfect $K_{a, b, c}$-tiling, that is, a spanning subgraph of $H$ consisting of vertex-disjoint copies of $K_{a, b, c}$. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for $r$-uniform hypergraphs for all $r\ge 3$. Our proof uses a lattice-ba…
Spanning Components and Surfaces Under Minimum Vertex Degree
Institut für Mathematik, Freie Universität Berlin（柏林自由大学数学研究所）；Institute for Basic Science（基础科学研究所）；Departament de Matemàtiques, Universitat Politècnica de Catalunya（加泰罗尼亚理工大学数学系）；Centre de Recerca Matemàtica（数学研究中心）；Insititut für Mathematik, Freie Universität Berlin（柏林自由大学数学研究所）
In this note we make some specific observations on the distribution of the degree of a given vertex in certain model of randomly growing networks. The rule for network growth is the following. Starting with an initial graph of minimum degree at least $k$, new vertices are added one by one. Each new vertex $v$ first chooses a random vertex $w$ to join to, where the probability of choosing $w$ is proportional to its degree. Then $k$ edges are added from $v$ to randomly chosen neighbours of $w$.
For a graph $G$, we denote by $σ_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $σ_{2}(G) = +\infty$. In this paper, we prove the following two results: (i) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a non-complete graph with $σ_{2}(G) \ge 2(s_{1} + s_{2} + 1) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $s_{i}+1$ with $σ_{2}(H_{i}) \ge 2s_{i} - 1$. (ii) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a triangle-free graph of order at least $3$ with $σ_{2}(G) \g…
We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration. Then we …
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result t…
A sequence $σ$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with respect to the class of polyhedral graphs, meaning that they admit exactly one realisation as a polyhedron. We focus on the case of sequences with largest entry $p-2$. We give a classification of polyhedral unigraphic sequences starting with $p-2,p-2$, as well as those starting with $p-2$ and containing exactly one $3$. Moreover, we characterise the unigraphic seque…
The Turan bound is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro-Wei inequality, which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We show how these statements can be generalized to directed graphs, thus yielding a bound on directed feedback vertex number in terms of vertex outdegrees and in terms of average outdegree, respectively.
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles …
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.
Answering a question of Häggkvist and Scott, Verstraëte proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $ε>0$ and large $k\geq k_ε$, average degree at least $k^2+3k-2+ε$ suffices, which is asymptotically tight for infinitely many graphs.
A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if…
We consider the problem of sampling a bipartite graph with given vertex degrees where a set $F$ of edges and non-edges which need to be contained is predefined. Our general result shows that the repeated swap of edges and non-edges in alternating cycles of at most size $2\ell-2$ ('$j$-swaps' with $j \leq 2 \ell-2$) in a current graph lead to an ergodic Metropolis Markov chain whenever $F$ does not contain a cycle of length $2 \ell$ with $\ell \geq 4.$ This leads to useful Markov chains whenever $\ell$ is not too large. If $F$ is a forest, $4$- and $6$-swaps are sufficient. Furthermore, we prov…
Let $G$ be a connected finite graph with vertex set $V(G)$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is defined as $\frac{1}{|V(G)|}\sum_{v \in V(G)}e(v)$. We show that the average eccentricity of a connected graph of order $n$, minimum degree $δ$ and maximum degree $Δ$ does not exceed $\frac{9}{4} \frac{n-Δ-1}{δ+1} \big( 1 + \frac{Δ-δ}{3n} \big) + 7$, and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing a $4$-cycles.
Given a graph $G$, a function $c:V(G)\longrightarrow \{1,\ldots,k\}$ with the property that $c(u)=c(v)=i$ implies that the distance between $u$ and $v$ is greater than $i$, is called a $k$-packing coloring of $G$. The smallest integer $k$ for which there exists a $k$-packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ$. Packing chromatic vertex-critical graphs are the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. A graph $G$ is called a packing chromatic critical graph if for every proper subgraph $H$ of $G$, $χ_ρ(H) < χ_ρ…
