tailieunhanh - Báo cáo toán học: " How to find overfull subgraphs in graphs with large maximum degree, II"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: How to find overfull subgraphs in graphs with large maximum degree, II. | How to find overfull subgraphs in graphs with large maximum degree II Thomas Niessen Institute of Statistics RWTH Aachen 52056 Aachen Germany niessen@ Submitted November 9 1999 Accepted December 10 2000. Mathematical Subject Classification 05C85. Abstract Let G be a simple graph with 3A G E . The Overfull Graph Conjecture states that the chromatic index of G is equal to A G if G does not contain an induced overfull subgraph H with A H A G and otherwise it is equal to A G 1. We present an algorithm that determines these subgraphs in O n5 3m time in general and in O n3 time if G is regular. Moreover it is shown that G can have at most three of these subgraphs. If 2A G E then G contains at most one of these subgraphs and our former algorithm for this situation is improved to run in linear time. 1 Introduction Let V G E G A G and X G denote the vertex set edge set maximum degree and chromatic index of a simple graph G respectively. In unambiguous cases we prefer to write V E A and x . G is called Class 1 if x A holds and otherwise G is called Class 2. By Vizing s Theorem 10 x A 1 holds for every Class 2 graph G. G is called overfull if IE V 2jA. Every overfull graph is Class 2 as well as every graph G having an overfull subgraph H with A H A G . We call such a subgraph A-overfull. It is easy to see that a subgraph H of G is A-overfull if and only if E H LỊV H 2JA G . Holyer 6 proved that the problem of deciding whether a graph is Class 1 is NP-complete. However for graphs with large maximum degree this problem seems to be easier. Chetwynd and Hilton 2 conjectured that a graph G with 2A V is Class 2 if and only if it has a A-overfull subgraph. This conjecture was known to be true for many THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R7 1 special cases when we presented an algorithm finding all induced A-overfull subgraphs of a graph G with 2A VI 8 . Recently Perkovic and Reed 9 proved that regular graphs of even order satisfying 2 A VI are .