The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that n 2 + ∗1 − 1 rb(G) n− ∗0 + 1, where ∗0 and ∗1 denote the independence number and the edge independence number, respectively, of th...

We provide an algorithm that converts any instance of the Hamiltonian cycle problem (HCP) into a cubic instance of HCP (3HCP), and prove that the input size of the new instance is only a linear function of that of the original instance. This result is reminiscent of the famous SAT to 3SAT conversion by Karp in 1972. Known conversions from directed HCP to undirected HCP, and sub-cubic HCP to cub...

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well known that a(G) ≤ 3 for any planar graph G, and that a(G) ≤ 2 for any planar graph G of diameter at most 2. The conjecture that every planar graph G without 3-cycles has a vertex partition (V1, V2) such that V1 is an ind...

The incidence coloring conjecture, proposed by Brualdi and Massey in 1993, states that the incidence coloring number of every graph is at most ∆ + 2, where ∆ is the maximum degree of a graph. The conjecture was shown to be false in general by Guiduli in 1997, following the work of Algor and Alon. However, in 2005 Maydanskiy proved that the conjecture holds for any graph with ∆ ≤ 3. It is easily...

A (1, 2)-eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of4$Y-operations. As a byproduct of the proof of t...

We consider cubic graphs formed with k ≥ 2 disjoint claws Ci ∼ K1,3 (0 ≤ i ≤ k−1) such that for every integer i modulo k the three vertices of degree 1 of Ci are joined to the three vertices of degree 1 of Ci−1 and joined to the three vertices of degree 1 of Ci+1. Denote by ti the vertex of degree 3 of Ci and by T the set {t1, t2, ..., tk−1}. In such a way we construct three distinct graphs, na...

Yutsis graphs are connected simple graphs which can be partitioned into two vertex-induced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus. Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, where they are used to generate summation formulae for general recoupling coefficients. Large Yutsis...

In this paper, a complete classification of finite simple cubic vertex-transitive graphs girth 6 is obtained. It proved that every such graph, with the exception Desargues graph on 20 vertices, either skeleton hexagonal tiling torus, truncation an arc-transitive triangulation closed hyperbolic surface, or 6-regular respect to dihedral scheme. Cubic larger than are also discussed.

We consider cubic graphs formed with k ≥ 2 disjoint claws Ci ∼ K1,3 (0 ≤ i ≤ k−1) such that for every integer i modulo k the three vertices of degree 1 of Ci are joined to the three vertices of degree 1 of Ci−1 and joined to the three vertices of degree 1 of Ci+1. Denote by ti the vertex of degree 3 of Ci and by T the set {t1, t2, ..., tk−1}. In such a way we construct three distinct graphs, na...

A graph is 1-regular if its automorphism group acts regularly on the set of its arcs. Miller [J. Comb. Theory, B, 10 (1971), 163–182] constructed an infinite family of cubic 1-regular graphs of order 2p, where p ≥ 13 is a prime congruent to 1 modulo 3. Marušič and Xu [J. Graph Theory, 25 (1997), 133– 138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs wit...