Contractible edges and liftable vertices in a 4-connected graph

Abstract

Let G be a 4-connected graph. An edge of G is said to be 4-contractible if the contraction of it results in a 4-connected graph. Let x be a vertex of G having degree 4. We denote the neighborhood of x by NG(x). We define an operation on G as follows; (1) Delete x from G, and (2) Add a perfect matching on NG(x). We call this operation “lifting” on x. Note that, in the lifting, we replace each resulting pair of double edges by a simple edge. A lifting with no resulting pair of double edges is said to be proper. A vertex x of G is said to be properly 4-liftable if there is a proper lifting on x which results in a 4-connected graph. Let Ec(G) and L(G) denote the set of 4-contractible edges of G and the set of properly 4-liftable vertices of G, respectively. Let U(0)(G) be the set of vertices of G having degree 4 which are incident with no contractible edges. By virtue of Saito's study [6], we obtain the following remarkable relation between 4-contractible edges and properly 4-liftable vertices of a 4-connected graph.Let G be a 4-connected graph of order at least six. Then,U(0)(G)⊆L(G).Using this result, we prove that2|Ec(G)|+|L(G)|≥max⁡{|V(G)|−|W(G)|,2|W(G)|}, where W(G) stands for the set of vertices of G having degree at least 5. We also show the sharpness of the bound.