A note on the dominating circuit conjecture and subgraphs of essentially 4-edge-connected cubic graphs

The well-known dominating circuit Conjecture has several interesting reformulations, for example conjectures of Fleischner, Matthews and Sumner, and Thomassen. We present another equivalent version of the dominating circuit conjecture considering subgraphs of essentially 4-edge-connected Cubic graphs. Let S = {u(1), u(2), u(3), u(4)} be a set of four distinct vertices of a graph G and V-2(G) be a set of all vertices of degree 2 of a graph G. We say that G is S-strongly dominating if the graph arising from G after adding two new edges e(1) = xy and e(2) = wz such that {x, y, w, z} = S has a dominating closed trail containing e(1) and e(2). We show that the dominating circuit conjecture is equivalent to the statement that any subgraph H of an essentially 4-edge-connectcd Cubic graph with |V-2(H)| = 4 and minimum degree delta(H) = 2 is strongly V-2(H)-dominating. (C) 2007 Elsevier B.V. All rights reserved.