A better bound for the cop number of general graphs

In this note, we prove that the cop number of any n‐vertex graph G, denoted by , is at most . Meyniel conjectured . It appears that the best previously known sublinear upper‐bound is due to Frankl, who proved . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 45–48, 2008     

On Meyniel's conjecture of the cop number

Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper‐bound O(n/ lnn) due to Chiniforooshan.     

On the structure of extremal graphs of high girth

Let n ≥ 3 be a positive integer, and let G be a simple graph of order v containing no cycles of length smaller than n + 1 and having the greatest possible number of edges (an extremal graph). Does G contain an n + 1-cycle? In this paper we establish some properties of extremal graphs and present several results where this question is answered affirmatively. For example, this is always the case for (i) v ≥ 8 and n = 5, or (ii) when v is large compared to n: v ≥ , where a = n - 3 - , n ≥ 12. On the other hand we prove that the answer to the question is negative for v = 2n + 2 ≥ 26. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 147–153, 1997     

On the girth of random Cayley graphs

We prove that random d‐regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_d‐1|G|)^1/2/2 and that random d‐regular Cayley graphs of simple algebraic groups over ??_q asymptotically almost surely have girth at least log _d‐1|G|/dim(G). For the symmetric p‐groups the girth is between loglog |G| and (log |G|)^α with α < 1. Several conjectures and open questions are presented. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.     

The circular chromatic number of series‐parallel graphs with large girth

It was proved by Hell and Zhu that, if G is a series‐parallel graph of girth at least 2⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). In this article, we prove that the girth requirement is sharp, i.e., for any k ≥ 2, there is a series‐parallel graph G of girth 2⌊(3k − 1)/2⌋ − 1 such that χ_c(G) > 4k/(2k − 1). © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 185–198, 2000     

Matchings in graphs of odd regularity and girth

We establish lower bounds on the matching number of graphs of given odd regularity d$d$ and odd girth g$g$, which are sharp for many values of d$d$ and g$g$. For d=g=5$d=g=5$, we characterize all extremal graphs.     

Variations on cops and robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≥1 edges at a time, establishing a general upper bound of , where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. We also show that in this case, the cop number of an n‐vertex graph can be as large as n^{1 ? 1/(R ? 2)} for finite R≥5, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)²/logn). Our approach is based on expansion. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

The Erd s-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erd s-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.     

3-restricted connectivity of graphs with given girth

Let G = （V, E） be a connected graph. X belong to V（G） is a vertex set. X is a 3-restricted cut of G, if G- X is not connected and every component of G- X has at least three vertices. The 3-restricted connectivity κ3（G） （in short κ3） of G is the cardinality of a minimum 3-restricted cut of G. X is called κ3-cut, if ｜X｜ = κ3. A graph G is κ3-connected, if a 3-restricted cut exists. Let G be a graph girth g ≥ 4, κ3（G） is min{d（x） ＋ d（y） ＋ d（z） - 4 ： xyz is a 2-path of G}. It will be shown that κ3（G） = ξ3（G） under the condition of girth.     

Injective coloring of planar graphs with girth 5

A coloring of a graph G is injective if its restriction to the neighbour of any vertex is injective. The injective chromatic number X_i\left(G\right) of a graph G is the leastk such that there is an injective k-coloring. In this paper, we prove that for each planar graph with g\ge 5 and \Delta \left(G\right)\ge 20, {\chi }_i\left(G\right)\le \Delta \left(G\right)+3.     

Note on robust critical graphs with large odd girth

A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e∈E(G). In this paper we show that for any integers k≥3 and l≥5 there exists a constant c=c(k,l)>0, such that for all , there exists a (k+1)-critical graph G on n vertices with and odd girth at least ?, which can be made (k−1)-colourable only by the omission of at least cn² edges.     

Flexibility of planar graphs of girth at least six

Let $G$ be a planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if $G$ has girth at least six and all lists have size at least three, then there exists an $L$-coloring respecting at least a constant fraction of the preferences.     

A lower bound on the order of regular graphs with given girth pair

The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 ( 1 ), 209–218]. A (δ, g)‐cage is a smallest δ‐regular graph with girth g. For all δ ≥ 3 and odd girth g ≥ 5, Harary and Kovács conjectured the existence of a (δ,g)‐cage that contains a cycle of length g + 1. In the main theorem of this article we present a lower bound on the order of a δ‐regular graph with odd girth g ≥ 5 and even girth h ≥ g + 3. We use this bound to show that every (δ,g)‐cage with δ ≥ 3 and g ∈ {5,7} contains a cycle of length g + 1, a result that can be seen as an extension of the aforementioned conjecture by Harary and Kovács for these values of δ, g. Moreover, for every odd g ≥ 5, we prove that the even girth of all (δ,g)‐cages with δ large enough is at most (3g ? 3)/2. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 153–163, 2007     

Decomposing graphs with girth at least five under degree constraints

We prove that the vertex set of a simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2. © 2000 John Wiley & Sons, Inc: J Graph Theory 33: 237–239, 2000     

The total chromatic number of regular graphs of high degree

The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) 32 |V (G)| + 263 , where d(G) denotes the degree of a vertex in G, then χT (G) d(G) + 2.