On the anti-Kekul number and odd cycle transversal of regular graphs
Let G G be a graph with a perfect matching. The anti-Kekul number is the smallest number of edges of G G whose removal will result in a connected subgraph with no perfect matching. In this note, we show that, if G G is a cyclically (r+1) ( r + 1 ) -edge-connected r r -regular graph (r?3 r ? 3 ) of even order, then either the anti-Kekul number of G G is at least r+1 r + 1 , or G G is not bipartite, and the smallest odd cycle transversal of G G has at most r r edges. For cubic graphs, the bound of the anti-Kekul number is optimal. Our result generalizes previous results on anti-Kekul numbers of fullerenes obtained by Kutnar et al. [K. Kutnar, J. Sedlar, D. Vuki?evi?, On the anti-Kekul number of leapfrog fullerenes, J. Math. Chem. 45 (2009) 431441.] and Yang et al. [Q. Yang, D. Ye, H. Zhang, Y. Lin, On the anti-Kekul number of fullerenes, MATCH Commun. Math. Comput. Chem. 67 (2) (2012) 281288].
anti-kekul number odd cycle transversal perfect matchings