The Relationship Between an Edge Colouring Conjecture and A Total Colouring Conjecture

Chetwynd and Hilton made the following \({edge-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) > \frac{1}{3}|V(G)|\), then \(G\) is Class \(2\) if and only if it contains an overfull subgraph \(H\) with \(\Delta(H) = \Delta(G)\). They also made the following \({total-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) \geq \frac{1}{2}(|V(G)|+ 1)\), then \(G\) is Type \(2\) if and only if \(G\) contains a non-conformable subgraph \(H\) with \(\Delta(H) = \Delta(G)\). Here we show that if the edge-colouring conjecture is true for graphs of even order satisfying \(\Delta(G) > \frac{1}{2}|V(G)|\), then the total-colouring conjecture is true for graphs of odd order satisfying \(\delta(G) \geq \frac{3}{4}{|V(G)|} – \frac{1}{4}\) and \(\text{def}(G) \geq 2(\Delta(G) – \delta(G) + 1)\).