K4‐free and ‐free Planar Matching Covered Graphs

Abstract

AbstractA bi‐subdivision of a graph J is a graph H obtained from J by subdividing each of its edges by inserting an even number of vertices. A matching covered subgraph H of a matching covered graph G is conformal if has a perfect matching. Using the theory of ear decompositions, Lovász (Combinatorica, 3 (1983), 105–117) showed that every nonbipartite matching covered graph has a conformal subgraph which is either a bi‐subdivision of K4 or of . (The graph is the triangular prism.) A matching covered graph is K4‐based if it contains a bi‐subdivision of K4 as a conformal subgraph; otherwise it is K4‐free. ‐based and ‐free graphs are analogously defined. The result of Lovász quoted above implies that any nonbipartite matching covered graph is either K4‐based or ‐based (or both). The problem of deciding which matching covered graphs are K4‐based and which are ‐based is, in general, unsolved. In this paper, we present a solution to this classification problem in the special case of planar graphs. In Section 2, we show that a matching covered graph is K4‐free (‐free) if and only if each of its bricks is K4‐free (‐free). In Section 5, we show that a planar brick is K4‐free if and only if it has precisely two odd faces. In Section 6, we determine the list of all ‐free planar bricks; apart from one exception, it consists of two infinite families of bricks. The principal tool we use for proving our results is the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769–817).