Overfull conjecture for graphs with high minimum degree

Abstract

Chetwynd and Hilton showed that any regular graph G of even order n which has relatively high degree Á(G ) ! (( ffiffiffi 7 p À 1)=2) n has a 1-factorization. This is equivalent to saying that under these conditions G has chromatic index equal to its maximum degree Á(G ). Using this result, we show that any (not necessarily regular) graph G of even order n that has sufficiently high minimum degree (G ) ! ( ffiffiffi 7 p =3) n has chromatic index equal to its maximum degree providing that G does not contain an ''overfull'' subgraph, that is, a subgraph which trivially forces the chromatic index to be more than the maximum degree. This result thus verifies the Overfull Conjecture for graphs of even order and sufficiently high minimum degree. ß

Chromatic index of dense quasirandom graphs

“…It was confirmed only for graphs with ∆(G) ≥ |V (G)| − 3 by Chetwynd and Hilton [3] in 1989. By restricting the minimum degree, Plantholt [15] in 2004 showed that the overfull conjecture is affirmative for graphs G with even order n and minimum degree δ ≥ √ 7n/3 ≈ 0.8819n. The 1-factorization conjecture is a special case of the overfull conjecture, which in 2013 was confirmed for large graphs by Csaba, Kühn, Lo, Osthus and Treglown [4].…”

Overfullness of critical class 2 graphs with a small core degree

“…Chetwynd and Hilton in 1986 [2,3] made a much stronger conjecture, commonly referred to as the Overfull Conjecture that for a ∆-critical graph of order n, if ∆(G) > n/3 then G is overfull. Except some very special results [3,8,11], the Overfull Conjecture seems untouchable with current edge-coloring techniques.…”

The overfullness of graphs with small minimum degree and large maximum degree

“…It was confirmed only for graphs with ∆ ≥ n − 3 by Chetwynd and Hilton [6] in 1989. By restricting the minimum degree, Plantholt [12] in 2004 showed that the overfull conjecture is affirmative for graphs G with even order n and minimum degree δ ≥ √ 7n/3 ≈ 0.8819n. The 1-factorization conjecture is a special case of the overfull conjecture (when G is regular with ∆ ≥ n/2), which in 2013 was confirmed for large n by Csaba, Kühn, Lo, Osthus, Treglown [7].…”