Decomposition of 3-connected cubic graphs

Decomposition of 3-connected graphs

Nonhamiltonian 3-Connected Cubic Planar Graphs

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vert...

Packing 3-vertex Paths in Cubic 3-connected Graphs

Let v(G) and λ(G) be the number of vertices and the maximum number of disjoint 3-vertex paths in G, respectively. We discuss the following old Problem. Is the following claim true ? (P ) if G is a 3-connected and cubic graph, then λ(G) = bv(G)/3c. We show, in particular, that claim (P ) is equivalent to some seemingly stronger claims (see 3.1). It follows that if claim (P ) is true, then Reed’s...

Small cycle covers of 3-connected cubic graphs

Every 2-connected simple cubic graph of order n ≥ 6 has a cycle cover with at most dn4 e cycles. This bound is best possible.

Hamiltonicity of Cubic 3-Connected k-Halin Graphs

We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let H = T ∪ C be a Halin graph, T being a tree and C the outer cycle. A k-Halin graph G can be obtained from H by adding edges while keeping planarity, joining vertices of H − C, such that G − C has at most k cycles. We prove that, in the class of cubic 3-connected graphs, all 14-Halin gr...