On the 2-extendability of planar graphs

AbstractSome sufficient conditions for the 2-extendability of k-connected k-regular (k⩾3) planar graphs are given. In particular, it is proved that for k⩾3, a k-connected k-regular planar graph with each cyclic cutset of sufficiently large size is 2-extendable

Extremal and degree donditions for path extendability in digraphs

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles

Characterizing minimally n-extendable bipartite graphs

AbstractIn this paper, it is proved that let G be a bipartite graph with bipartition (X,Y) and with a perfect matching M, let G be an n-extendable graph, then G is minimally n-extendable if and only if, for any two vertices x∈X and y∈Y such that xy∈E(G), there are exactly n internally disjoint (x,y)M-alternating paths P1,P2,…,Pn such that Pi(1⩽i⩽n) starts and ends with edges in E(G)⧹M