The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer μ0(g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value μ0(g) is attained by a graph whose girth is exactly g. The values of μ0(g) when 3 ≤ g ≤ 8 have been known for over thirty y...

The circumference c(G) of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5and 6-edge-connected cubic graphs with circumference ratio c(G)/|V (G)| bounded from above by 0.876, 0.960 and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a c...

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

Abstract A connected graph of order n admitting a semiregular automorphism / k is called -multicirculant. Highly symmetric multicirculants small valency have been extensively studied, and several classification results exist for cubic vertex- arc-transitive multicirculants. In this paper, we study the broader class vertex-transitive graphs an /3 or larger that may not be semiregular. particular...