We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.12776. The lower bound on the si...

An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-c...

continue to investigate the problem of the existence of a Hamilton connected cubic (m,n)-metacirculant graphs. We show that a connected cubic (m,n)-metacirculant graph G = MC(m, n, a, So, ,.., has Hamilton cycle if either a 2 == 1 (mod n) or in the case of an odd number f.L one of the numbers (a 1) or a + a 2 _ ... ajJ.-2 + ajJ.-l) relatively to n. As a corollary of results we obtain that every...

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.

We study the equivalence problem of cubic forms. We lower bound its complexity by that of F-algebra isomorphism problem and hence by the graph isomorphism problem (for all fields F). For finite fields we upper bound the complexity of cubic forms by NP∩coAM. We also study the cubic forms obtained from F-algebras and show that they are regular and indecomposable.

We prove that a random cubic graph almost surely is not homomorphic to a cycle of size 7. This implies that there exist cubic graphs of arbitrarily high girth with no homomorphisms to the cycle of size 7.

A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive. It is proved that the Gray graph is the only cubic semisymmetric graph of order 2p3, where p 3 is a prime.

A regular graph is called semisymmetric if it is edge transitive but not vertex transitive It is proved that the Gray graph is the only cubic semisymmetric graph of order p where p is a prime

In 1993, Brualdi and Massey conjectured that every graph can be incidence colored with ∆+2 colors, where ∆ is the maximum degree of a graph. Although this conjecture was solved in the negative by an example in [1], it might hold for some special classes of graphs. In this paper, we consider graphs with maximum degree ∆ = 3 and show that the conjecture holds for cubic Hamiltonian graphs and some...

A well-known conjecture of Lovász and Plummer from the mid-1970’s, still open, asserts that for every cubic graph G with no cutedge, the number of perfect matchings in G is exponential in |V (G)|. In this paper we prove the conjecture for planar graphs; we prove that if G is a planar cubic graph with no cutedge, then G has at least 2 (G)|/655978752