Perfect matching index of cubic graphs of defect 3
Roman Nedela

Abstract

The colouring defect of a bridgeless cubic graph G is the minimum number 
of edges left uncovered by a multiset of three perfect matchings of $G$. 
By the Petersen theorem the defect is well-defined for bridgeless cubic 
graphs. Clearly, G is 3-edge-colourable if and only if the defect of G 
is zero. Since in a snark any two perfect matchings have a non-empty 
intersection, the least possible non-zero defect is 3. The colouring 
defect was introduced and investigated in a serie of papers by Steffen 
2015 and Jin and Steffen 2017. Among others they proved that the defect 
is at least g(G)/2, where g(G) denotes the girth. Since Kochol 
constructed snarks of arbitrarily large girth, the defect is unbounded.

Motivated by the Berge-Fulkerson conjecture we investigate the perfect 
matching index of bridgeless cubic graphs of defect 3. Steffen in 2015 
proved that the perfect matching index of cyclically 4-connected cubic 
graphs of defect 3 is at most 5. We have extended his result in two ways 
by proving

Theorem A: The perfect matching index of a cyclically 4-connected cubic 
graph G of defect 3  is four unless G is the Petersen graph.

Theorem B: Every bridgeless cubic graph of defect 3 has a Berge cover.

Recall that the Berge cover of G is a system od at most five perfect 
matchings covering all edges of G and the perfect matching index is the 
minimum number of perfect matchings covering all edges.

This a joint work with J. Karabas, E. Macajova and M. Skoviera.