The Dominating Circuit Conjecture and Subgraphs of Essentially 4-Edge Connected Cubic Graphs

The well-known Dominating circuit conjecture has several interesting reformulation, for example conjectures of Fleischner, Matthews-Sumner and Thomassen. We present another equivalent version of the Dominating circuit conjecture considering subgraphs of essentially 4-edge-connected cubic graphs. = { 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 = x y and e 2 = w z such that { x , y , w , z } = S has a dominating circuit containing e 1 and e 2 . We show, that the dominating circuit conjecture is equivalent with the statement that any subgraph H of essentially 4-edge-connected cubic graph with | V 2 ( H ) | = 4 and minimum degree δ ( H ) = 2 is strongly V 2 ( H ) -dominating.