Cycle covers (I) – Minimal contra pairs and Hamilton weights

Abstract

Let G be a bridgeless cubic graph associated with an eulerian weight w : E ( G ) ↦ { 1 , 2 } . A faithful circuit cover of the pair ( G , w ) is a family of circuits in G which covers each edge e of G precisely w ( e ) times. A circuit C of G is removable if the graph obtained from G by deleting all weight 1 edges contained in C remains bridgeless. A pair ( G , w ) is called a contra pair if it has no faithful circuit cover, and a contra pair ( G , w ) is minimal if it has no removable circuit, but for each weight 2 edge e, the graph G − e has a faithful circuit cover with respect to the weight w. It is proved by Alspach et al. (1994) [2] that if ( G , w ) is a minimal contra pair, then the graph G must contain a Petersen minor. It is further conjectured by Fleischner and Jackson (1988) [5] that this graph G must be the Petersen graph itself (not just as a minor). In this paper, we prove that this conjecture is true if every Hamilton weight graph is constructed from K 4 via a series of ( Y → △ ) -operations.