We introduce weak oddness ωw, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph G, %(G) ≤ ωw(G) ≤ ω(G), where %(G) denotes the resistance of G and ω(G) denotes the oddness of G, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs G satisfying...

Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system PG(n, 2), n ≥ 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater ...

It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The p...

Generalized recoupling coefficients or 3nj-coefficients can be expressed as multiple sums over products of Racah or 6j-coefficients [1]. The problem of finding an optimal summation formula (i.e. with a minimal number of Racah coefficients) for a given 3nj-coefficient is equivalent to finding an optimal reduction of a so-called Yutsis graph [2]. In terms of graph theory Yutsis graphs are connect...

A 1-factor M of a cubic graph G is strong if |M ∩ T |= 1 for each 3-edge-cut T of G. It is proved in this paper that a cubic graph G has precisely three strong 1-factors if and only if the graph can be obtained from K4 via a series of 4 ↔ Y operations. Consequently, the graph G admits a Hamilton weight and is uniquely edge-3-colorable. c © 2001 Elsevier Science B.V. All rights reserved.

A perfectly one-factorable (P1F) regular graph G is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. We consider the case in which G is a cubic graph. The existence of a P1F cubic graph is guaranteed for each admissible value of the number of vertices. We give conditions for determining P1F graphs within a subfamily of...

For k = 2, 3 and a cubic graph G let νk(G) denote the size of a maximum k-edge-colorable subgraph of G. Mkrtchyan, Petrosyan and Vardanyan proved that ν2(G) ≥ 4 5 · |V (G)|, ν3(G) ≥ 7 6 · |V (G)| [13]. They were also able to show that ν2(G) + ν3(G) ≥ 2 · |V (G)| [3] and ν2(G) ≤ |V (G)|+2·ν3(G) 4 [13]. In the present work, we show that the last two inequalities imply the first two of them. Moreo...

In this paper, we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of k-edge-colourings of a connected k-regular graph on n vertices is k · ((k − 1)!). Our proof is constructive and leads to a branching algorithm enumerating all the k-edge-colourings of a connected k-regular graph in time O∗(((k − 1)!)) and pol...