Let G be a simple graph with maximum degree Δ(G). A subgraph H of is overfull if |E(H)|>Δ(G)⌊|V(H)|/2⌋. Chetwynd and Hilton in 1986 conjectured that on n vertices Δ(G)>n/3 has chromatic index Δ(G) only contains no subgraph. Glock, Kühn Osthus 2016 showed the conjecture true for dense quasirandom graphs even order, they same should hold such odd order. In this paper, we show affirmative.

The Wiener index of a graph W(G) is well studied topological for graphs. An outstanding problem Šoltés to find graphs G such that W(G)=W(G−v) all vertices v∈V(G), with the only known example being G=C11. We relax this by defining notion indices signed graphs, which we denote Wσ(G), and under relaxation construct many Wσ(G)=Wσ(G−v) v∈V(G). This ends up related independent interest, asks when it ...