Extendability of the complementary prism of bipartite graphs

For a nonnegative integer k, a connected graph G of order at least 2k+2 is k-extendable if G has a perfect matching and every set of k independent edges extends to a perfect matching in G. The largest integer k such that G is k-extendable is called the extendability of G. The complementary prism GG of G is the graph constructed from G and its complement G defined on a set of vertices disjoint from V (G) (i.e. V (G) ∩ V (G) = ∅) by joining each pair of corresponding vertices by an edge. Janseana and ∗ N.A.: Also at Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand. Work supported by the Thailand Research Fund grant #BRG5480014. † W.A.: Work supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research(C), 25330017, 2013–2015 and Symposium Promotion Grant of Institute of Natural Sciences at Nihon University for 2014. ‡ A.S.: Work supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research(C), 25330017, 2013–2015. N. ANANCHUEN ET AL. /AUSTRALAS. J. COMBIN. 66 (3) (2016), 436–448 437 Ananchuen [Thai J. Math. 13 (2015), 703–721] gave a lower bound to the extendability of GG in terms of the extendabilities of G and G in the case that neither G nor G is a bipartite graph. In this paper, we consider the remaining case and give a sharp lower bound to the extendability of GG when G is a bipartite graph.