We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [8] and Bondy and Locke [2] proved that every (sub)cubic graph of girth at least 4 has a...

Let k ≥ 3 be a positive odd integer and q be a power of a prime. In this paper we give an explicit construction of a q–regular bipartite graph on v = 2q vertices with girth g ≥ k + 5. The constructed graph is the incidence graph of a flag–transitive semiplane. For any positive integer t we also give an example of a q = 2–regular bipartite graph on v = 2q vertices with girth g ≥ k + 5 which is b...

We consider the complexity of oriented homomorphism and two of its variants, namely strong oriented homomorphism and pushable homomorphism, for planar graphs with large girth. In each case, we consider the smallest target graph such that the corresponding homomorphism is NP-complete. These target graphs T4, T5, and T6 have 4, 5, and 6 vertices, respectively. For i ∈ {4, 5, 6} and for every g, w...

Let C2k denote the cycle of length 2k and Cg = {C3, C4, . . . , Cg}. The girth of a graph G containing a cycle is the length of a shortest cycle in G, so a graph has girth at least g + 1 if it is Cg-free. The distance between vertices u and v in a graph G, denoted dG(u, v) is the minimum length of a uv-path in G. Then dG(·, ·) is a metric on V (G). The diameter of a connected graph G is max{dG(...

A graph is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a {d, d + 1}semiregular graph if all its vertices have degree either d or d+1. A smallest {d, d+1}-semiregular graph G with girth g is said to be a ({d, d+1}; g)-cage. We show that every ({d, d+1}; g)-cage with odd girth g is edge-supe...

Motivated by the work of Nešetřil and Rödl on “Partitions of vertices”, we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r-coloring of the vertices of H yields a monochromatic...

For a finite, simple, undirected graph G and an integer d ≥ 1, a mindeg-d subgraph is a subgraph of G of minimum degree at least d. The dgirth of G, denoted gd(G), is the minimum size of a mindeg-d subgraph of G. It is a natural generalization of the usual girth, which coincides with the 2-girth. The notion of d-girth was proposed by Erdős et al. [13, 14] and Bollobás and Brightwell [7] over 20...