The 2-extendability of 5-connected graphs on the Klein bottle

Seiya Negami, Yusuke Suzuki
2010 Discrete Mathematics  
A graph is said to be k-extendable if any independent set of k edges extends to a perfect matching. In this paper, we shall characterize the forbidden structures for 5-connected graphs on the Klein bottle to be 2-extendable. This fact also gives us a sharp lower bound of representativity of 5-connected graphs embedded on the Klein bottle to have such a property, which was considered in Kawarabayashi et al. (submitted for publication) [4] . A graph in this paper is a simple graph, that is, one
more » ... th no loops and no multiple edges. We denote the vertex set and the edge set of a graph G by V (G) and E(G), respectively. The number of vertices of G is often called the order of G. A set M of edges in a graph G is said to be a matching (or o members of M share a vertex. A matching M is perfect if every vertex of G is covered by an edge of M. A graph G with |V (G)| ≥ 2k + 2 is said to be k-extendable if every matching M ⊆ E(G) with |M| = k, extends to a perfect matching in G. Plummer [8, 9, 7] has introduced this notion of k-extendability of graphs and discussed it, combining topological properties. For example in [9], he has proved that every 5-connected planar graph of even order is 2-extendable. Let G be a graph and {e 1 , e 2 } an independent pair of edges e 1 = u 1 v 1 and e 2 = u 2 v 2 . If G − {u 1 , v 1 , u 2 , v 2 } has an odd component, that is, a connected component consisting of an odd number of vertices, then the subgraph in G induced by the odd component and {u 1 , v 1 , u 2 , v 2 } is called a generalized butterfly. It is clear that if G contains a generalized butterfly, then G is not 2-extendable; any matching containing the two edges e 1 and e 2 cannot cover all vertices in the odd component. By these facts, Plummer proved the following theorem: Theorem 1 ([9]). Every 4-connected maximal planar graph of even order is 2-extendable unless it contains a generalized butterfly. More generally, Plummer [7] has shown that for a given closed surface, there exists an upper bound for a natural number k such that the surface admits embeddings of k-extendable graphs and Dean [3] has determined the precise value of the bound. Recently, Aldred et al. [1] have proven that a triangulation of even order on a closed surface of positive genus is 2-extendable if it has sufficiently large representativity. (The representativity of G on a closed surface F 2 denoted by γ (G) is defined as follows: γ (G) := min{|G ∩ | : is an essential simple closed curve on F 2 }. A graph G on F 2 is said to be r-representative if γ (G) ≥ r.) Furthermore in [6], Mizukai et al. have discussed the 2-extendability of 5-connected graphs on the torus *
doi:10.1016/j.disc.2010.06.020 fatcat:nojktoeqnjfj7mjd5umtqxot5y