1‐factorization of regular graphs by colour exchange

Abstract

AbstractWe present a new general theory that deals with the problem of determining a 1‐factorization of a graph using only the elementary technique of colour exchange. Our work is inspired by an old question of Vizing, who in [Cybernetics 3 (1965) 32–41] asked whether an optimal edge colouring of any multigraph G can always be obtained from an arbitrary edge colouring of G by repeatedly exchanging colours along bicoloured chains and suppressing empty colour classes. We conjecture that the answer to Vizing's question is affirmative. We apply our theory to the class of regular graphs of even order at most 10, proving the validity of this conjecture for this class of graphs. This yields an algorithm for finding a 1‐factorization of any 1‐factorizable graph of order at most 10. We also formulate two (stronger) conjectures and prove that they hold for the same class of graphs. Our method can be extended to graphs of larger order and to non‐regular graphs or multigraphs.