Non-conformable subgraphs of non-conformable graphs Original Research Article

We show that if G and H are non-conformable graphs, with H being a subgraph of G of the same maximum degree Δ(G), and if Δ(G)⩾⌈12|V(G)|⌉, then |V(H)|=|V(G)|. We also show that this inequality is best possible, for when Δ(G)=⌊12|V(G)|⌋ there are examples of graphs G and H with Δ(H)=Δ(G) and |V(H)|<|V(G)| which are both non-conformable. We determine all such examples. Interest in this stems from the modified Conformability Conjecture of Chetwynd, Hilton and Hind, which would characterize all graphs G with Δ(G)⩾⌈12|V(G)|⌉, for which the total chromatic number χT(G) satisfies χT(G)=Δ(G)+1, in terms of non-conformable subgraphs.