The Matching Extendability of Optimal 1-Planar Graphs
A graph G is said to be 1-planar if it can be drawn on the sphere or plane so that any edge of G has at most one crossing point with another edge. Moreover, G is called an optimal 1-planar graph if \(|E(G)| = 4|V(G)|-8\). In this paper, we investigate the matching extendability of optimal 1-planar graphs. It is shown that every optimal 1-planar graph G of even order is 2-extendable unless G contains a 4-cycle C which separates the graph into two odd components. Moreover, for any 5-connected optimal 1-planar graph, we characterize a matching with three edges which is not extendable.
KeywordsOptimal 1-planar graph Perfect matching Extendability
The authors would like to thank Katsuhiro Ota whose comment led to significant improvement in the proof of Lemma 4.
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