A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ where $χ(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\ell P_1)$-free graphs for all $k$ and all $\ell$. Together with previous results, the only graphs $H$ for which it is unknown if there are an infinite number of $k$-vertex-critical $H$-free graphs is $H=(P_4+\ell P_1)$ for all $\ell\ge 1$. We consider a restriction on the smallest open case, and show that there are only finitely many $k$-vertex-critical (gem, co-gem)-free graphs for all $k$, where gem$=\overline{P_4+P_1}$. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of $C_5$. This characterization allows us to give the complete list of all $k$-vertex-critical (gem, co-gem)-free graphs for all $k\le 16$
For a graph $G$, let $χ(G)$ and $ω(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies $χ(G)\le \lceil\frac{5ω(G)}{4}\rceil$. Moreover, this bound is best possible.
We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$…
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 $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $k\ge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3…
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, \ell}(k)$ that meets all such minor-mod…
Let $G$ be a graph. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. A {\em diamond} is a graph obtained from two triangles that share exactly one edge. A {\em kite} is a graph consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. In this paper, we prove some structural properties to $(P_7, C_4,$ diamond)-free graphs, $(P_7, C_4,$ kite)-free graphs and $(P_7, C_4,$ gem)-free graphs. As their corollaries, we show that (\romann…
A \emph{$k$-tree} is a chordal graph with no $(k+2)$-clique. An \emph{$\ell$-tree-partition} of a graph $G$ is a vertex partition of $G$ into `bags', such that contracting each bag to a single vertex gives an $\ell$-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all $k\geq\ell\geq0$, every $k$-tree has an $\ell$-tree-partition in which every bag induces a connected $\floor{k/(\ell+1)}$-tree. An analogous result is proved for oriented $k$-trees.
For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or $P_3 + P_1$. Our results imply the existence of new polynomial-time certifying algorithms for deciding the $k$-colorability of $(P_2+\ell P…
The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs having no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). It is known that this fails when $n=2\ell$.
The Turán number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{C}_{\ell}^{\left(k \right)}$ denote the family of all $k$-uniform minimal cycles of length $\ell$, $\mathcal{S}(\ell_1,\ldots,\ell_r)$ denote the family of hypergraphs consisting of unions of $r$ vertex disjoint minimal cycles of length $\ell_1,\ldots,\ell_r$, respectively, and $\mathbb{C}_{\ell}^{\left(k \right)}$ denote a $k$-uniform linear cycle of length $\ell$. We d…
Vertex leaky forcing was recently introduced as a new variation of zero forcing in order to show how vertex leaks can disrupt the zero forcing process in a graph. An edge leak is an edge that is not allowed to be forced across during the zero forcing process. The $\ell$-edge-leaky forcing number of a graph is the size of a smallest zero forcing set that can force the graph blue despite $\ell$ edge leaks. This paper contains an analysis of the effect of edge leaks on the zero forcing process instead of vertex leaks. Furthermore, specified $\ell$-leaky forcing is introduced. The main result is t…
A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists a vertex $w\in V(G)$ such that $\ell(u)+\ell(v) = \ell(w)$. We say that sum labeling $\ell$ is minimal if there is a vertex $u\in V(G)$ such that $\ell(u)=1$. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the ques…
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 $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle $C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even o…
A (vertex) $\ell$-ranking is a colouring $φ:V(G)\to\mathbb{N}$ of the vertices of a graph $G$ with integer colours so that for any path $u_0,\ldots,u_p$ of length at most $\ell$, $φ(u_0)\neqφ(u_p)$ or $φ(u_0)<\max\{φ(u_0),\ldots,φ(u_p)\}$. We show that, for any fixed integer $\ell\ge 2$, every $n$-vertex planar graph has an $\ell$-ranking using $O(\log n/\log\log\log n)$ colours and this is tight even when $\ell=2$; for infinitely many values of $n$, there are $n$-vertex planar graphs, for which any 2-ranking requires $Ω(\log n/\log\log\log n)$ colours. This result also extends to bounded genu…
We study the problem of finding pairwise vertex-disjoint copies of the $\ell$-vertex cycle $C_\ell$ in the randomly perturbed graph model, which is the union of a deterministic $n$-vertex graph $G$ and the binomial random graph $G(n,p)$. For $\ell \ge 3$ we prove that asymptotically almost surely $G \cup G(n,p)$ contains $\min \{δ(G), \lfloor n/\ell \rfloor \}$ pairwise vertex-disjoint cycles $C_\ell$, provided $p \ge C \log n/n$ for $C$ sufficiently large. Moreover, when $δ(G) \geαn$ with $0<α\le 1/\ell$ and $G$ is not `close' to the complete bipartite graph $K_{αn,(1-α) n}$, then $p \ge C/n$…
For given integers $r$ and $\ell$ such that $2\leqslant\ell\leqslant r-1$, an $r$-uniform hypergraph $H$ is called a partial Steiner $(n,r,\ell)$-system, if every subset of size $\ell$ lies in at most one edge of $H$. In particular, partial Steiner $(n,r,2)$-systems are also called linear hypergraphs. The partial Steiner $(n,r,\ell)$-system process starts with an empty hypergraph on vertex set $[n]$ at time $0$, the $ \binom{n}{r}$ edges arrive one by one according to a uniformly chosen permutation, and each edge is added if and only if it does not overlap any of the previously-added edges in …
We prove that for every graph, any vertex subset $S$, and given integers $k,\ell$: there are $k$ disjoint cycles of length at least $\ell$ that each contain at least one vertex from $S$, or a vertex set of size $O(\ell \cdot k \log k)$ that meets all such cycles. This generalises previous results of Fiorini and Hendrickx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in $O(k \log k \cdot s^2 \cdot (f(\ell) \cdot n+m))$ time, where $s$ denotes the cardinality of $S$.
A tight $k$-uniform $\ell$-cycle, denoted by $TC_\ell^k$, is a $k$-uniform hypergraph whose vertex set is $v_0, \cdots, v_{\ell-1}$, and the edges are all the $k$-tuples $\{v_i, v_{i+1}, \cdots, v_{i+k-1}\}$, with subscripts modulo $\ell$. Motivated by a classic result in graph theory that every $n$-vertex cycle-free graph has at most $n-1$ edges, Sós and, independently, Verstraëte asked whether for every integer $k$, a $k$-uniform $n$-vertex hypergraph without any tight $k$-uniform cycles has at most $\binom{n-1}{k-1}$ edges. In this paper, we answer this question in negative.
The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu, Zhang and Li introduced the concept of equitable $(t,k)$-tree-coloring, which can be regarded as a generalization of proper equitable $t$-coloring. The \emph{strong equitable vertex $k$-arboricity} of $G$, denoted by ${va_k}^\equiv(G)$, is the smallest integer $t$ such that $G$ has an equitable $(t', k)$-tree-coloring for every $t'\geq t$. The exact value of strong equitable vertex $k$-arboricity of complete equipartition bipartite graph $K_{n,n}$ was studied by Wu, Zhan…
For any graph $F$ and any integer $r\geq 2$, the \emph{online vertex-Ramsey density of $F$ and $r$}, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs.\ Builder). This parameter was introduced in a recent paper \cite{mrs11}, where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs.\ the binomial random graph $G_{n,p}$). For a large class of graphs $F$, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple …
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with…
The Reconstruction Conjecture of Ulam asserts that, for $n\geq 3$, every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-1$ vertices. The conjecture is known to hold for various special classes of graphs but remains wide open. We survey results on the more general conjecture by Kelly from 1957 that for every positive integer $\ell$ there exists $M_\ell$ (with $M_1=3$) such that when $n\geq M_\ell$ every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-\ell$ vertices.
Given a graph $G$ and an integer $\ell\ge 2$, we denote by $α_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $α_{\ell}(G) = o(n)$, which can be seen as a Ramsey--Turán variant of the celebrated Hajnal--Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $α_\ell(G)=n^{1-o(1)}$ for all $r\ge \ell\ge 2$.
In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d, m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance $m$-tuple ($\ell, r$)-dominating set if it satisfies the following conditions: (i) each vertex $v \in V$ is $d$-distance dominated by at least $m$ vertices in $V'$, and (ii) each $r$ size subset $U$ of $V$ is $d$-distance dominated by at least $\ell$ vertices in $V'$. Here, a vertex $v$ is $d$-distance dominated by another vertex $u$ means the shortest path distance bet…
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow $\ell$-cycle $C_{\ell}$ whenever $n \ge 432 \ell$. This result is sharp for all odd integers $\ell \geq 3$, and extends earlier work of the authors for when $\ell$ is even.
A configuration of a graph is an assignment of one of two states, on or off, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an on vertex. The following result is proved in this note: given any starting configuration $x$ of a tree, if there is a sequence of regular moves which brings $x$ to another configuration in which there are $\ell$ on vertices then there must exist a sequence of valid moves which takes $x$ to a configuration with at most $\ell +2$ on vertices. We provide example to show that the upper …
The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree}…
For integers $k\ge 3$ and $1\le \ell\le k-1$, we prove that for any $α>0$, there exist $ε>0$ and $C>0$ such that for sufficiently large $n\in (k-\ell)\mathbb{N}$, the union of a $k$-uniform hypergraph with minimum vertex degree $αn^{k-1}$ and a binomial random $k$-uniform hypergraph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-(k-\ell)-ε}$ for $\ell\ge 2$ and $p\ge C n^{-(k-1)}$ for $\ell=1$ on the same vertex set contains a Hamiltonian $\ell$-cycle with high probability. Our result is best possible up to the values of $ε$ and $C$ and answers a question of Krivelevich, Kwan and Sudakov.
Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}-x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ \mathcal{A}_{G} $, and freeness of $ \mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ ψ$-graphical arrangement $ \mathcal{A}_{G, ψ} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ ψ$-graphical arrangemen…
The concept of the \emph{McPherson number} of a simple connected graph $G$ on $n$ vertices denoted by $Υ(G)$, is introduced. The recursive concept, called the \emph{McPherson recursion}, is a series of \emph{vertex explosions} such that on the first interation a vertex $v \in V(G)$ explodes to arc (directed edges) to all vertices $u \in V(G)$ for which the edge $vu \notin E(G)$, to obtain the mixed graph $G'_1.$ Now $G'_1$ is considered on the second iteration and a vertex $w \in V(G'_1) = V(G)$ may explode to arc to all vertices $z \in V(G'_1)$ if edge $wz \notin E(G)$ and arc $(w, z)$ or $(z…
Characterization of strongly $\mathbb{Z}_\ell$-connected graphs of small order
School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院和LPMC）；Department of Mathematical and Physical Sciences, Henan Institute of Engineering（河南工程学院数理学院）
Assume we are given a graph $G$, two independent sets $S$ and $T$ in $G$ of size $k \geq 1$, and a positive integer $\ell \geq 1$. The goal is to decide whether there exists a sequence $\langle I_0, I_1, ..., I_\ell \rangle$ of independent sets such that for all $j \in \{0,\ldots,\ell-1\}$ the set $I_j$ is an independent set of size $k$, $I_0 = S$, $I_\ell = T$, and $I_{j+1}$ is obtained from $I_j$ by a predetermined reconfiguration rule. We consider two reconfiguration rules. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Tok…
Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$ vertex-disjoint copies of $Y_{3,2}$, for $m\le n/7$. The bound on the number of edges is asymptotically best possible. This problem generalizes the Matching Conjecture of Erdős. We then use this result combined with the absorbing method to determine the asymptotically best possible minimum $(k-3)$-degree threshold for $\ell$-Hamiltonicity in $k$-graphs, whe…
The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $δ(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $α_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \e…
We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number $D_t(\ell)$ of vertices of each degree $\ell$ at each time $t$, focussing particularly on the case where $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ is concentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell \le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.
It is known that the spectral radius of a digraph with k edges is \le \sqrt{k}, and that this inequality is strict except when k is a perfect square. For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, and a corresponding loop, and then connecting it to the first \lfloor \ell/2\rfloor vertices by pairs of directed edges (this is for odd \ell, for even \ell we add one extra edge to the new vertex). Using a combinatorial reciprocity theorem by Gessel, and a classification by Backelin on t…
