A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2-regular (following the notation of Djokovic), or...

Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in 2 colors such that each monochromatic component is a p...

We show that for every integer d ∈ N, there is N(d) ∈ N such that if K is any field and F is a finite subset of GLd(K), which generates a non amenable subgroup, then F contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for non-amenable linear groups, and has cons...

A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every planar graph of girth 6 (respectively 7,8) can be star colored from lists of size 8 (respectively 7,6). We give an example of a planar graph of girth 5 that requires 6 colors to star color.

Consider the following problem: Does there exist a graph with large girth and large chromatic number? The girth of a graph is the length of the shortest cycle and the chromatic number is the minimum number of colors needed to color the vertices of the graph so no two adjacent vertices have the same color. These two conditions seem contradictory, intuitively to have high chromatic number we need...

Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H. Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan [11] proved that it is NP-complete to determine if a given graph G is the square of some graph. The graph introduced in their ...

We consider the problem of construction of graphs with given degree k and girth 5 and as few vertices as possible. We give a construction of a family of girth 5 graphs based on relative difference sets. This family contains the smallest known graph of degree 8 and girth 5 which was constructed by G. Royle, four of the known cages including the Hoffman-Singleton graph, some graphs constructed by...