On the clique-transversal number in (claw, K 4)-free 4-regular graphs

The clique-transversal number of a{K1,3,K4}-free 4-regular graph

“…Shan and Kang [14] give a lower bound on the clique-transversal number for claw-free 4-regular graphs and characterized the extremal graphs achieving the lower bound. Wang et al [16] proved that τ c (G) = ⌈ n 3 ⌉ for any 2-connected {K 1,3 , K 4 }-free 4-regular graph G of order n by induction on n. We give a short proof of this theorem. Further, they posed the following conjecture.…”

“…Based on Theorems 2.2 and 2.5, we give a short proof for the following theorem of Wang et al [16]. Theorem 2.6 (Wang, Shan and Liang [16]).…”

“…Theorem 2.6 (Wang, Shan and Liang [16]). Let G be a 2-connected {K 1,3 , K 4 }-free 4-regular graph of order n. Then τ c (G) = ⌈ n 3 ⌉.…”