A new bound for parsimonious edge-colouring of graphs with maximum degree three

In a graph G of maximum degree 3, let γ(G) denote the largest fraction of edges that can be 3 edge-coloured. Rizzi [9] showed that γ(G) ≥ 1 − 2 3godd(G) where godd(G) is the odd girth of G, when G is triangle-free. In [3] we extended that result to graph with maximum degree 3. We show here that γ(G) ≥ 1− 2 3godd(G)+2 , which leads to γ(G) ≥ 15 17 when considering graphs with odd girth at least 5, distinct from the Petersen graph.