Perfect Matchings Avoiding Several Independent Edges in a Star‐Free Graph

Abstract

Abstract:In Aldred and Plummer (Discrete Math 197/198 (1999) [29][30][31][32][33][34][35][36][37][38][39][40] proved that every m-connected K 1,m−2k−l +2 -free graph of even order has a perfect matching M with F 1 ⊆ M and M ∩ F 2 = ∅, where F 1 and F 2 are prescribed disjoint sets of independent edges with |F 1 | = k and, then the star-free condition in the above result is best possible. In this paper, forwe prove a refinement of the result in which the condition is replaced by the weaker condition that G is K 1, m−2k+2 2 -free (note that the new condition does not depend on l ). We also show that if m is even and either k ≥ 1 or l ≤ 3m 4 , then for m-connected graphs G with sufficiently large order, one can replace the condition by the still weaker condition that G is K 1, m−2k+4 2

Perfect matchings avoiding prescribed edges in a star-free graph

“…This makes us speculate that in this range, if we only specify independent edges to be avoided, the condition in Theorem 1 might be relaxed. Very recently, Egawa and Furuya [4] have verified this speculation by forbidding a star larger than K 1,m−k+2 to guarantee an m-connected graph to be E(0, k).…”

“…Therefore, we have to give a different proof in the range 4 ≤ m ≤ 6. However, in this range, the only possible values for (m, k) satisfying 1 2 (m + 2) ≤ k ≤ m − 1 and k ≡ 2m + 4 (mod 3) are (6,4) and (4,3). We give a proof specifically designed for these values.…”