Forbidden Pairs for Connected Even Factors in Supereulerian Graphs

A graph is called supereulerian if it has a spanning eulerian subgraph. A connected even [2, 2s]-factor of a graph G is a connected factor with all vertices of even degree i(i∈ { 2 , 4 , … , 2 s} ) , where s≥ 1 is an integer. Let W_s,t be the graph obtained from vertex-disjoint sK₂ and (t+ 1 ) K₁ by adding all possible edges between exactly one K₁ and the remaining graphs sK₂ and tK₁ . When s= t= 1 , W_{1 , 1} is Z₁ . In this paper, we show that for any positive integer k≤ 4 and 2 ≤ i≤ k+ 1 , every connected supereulerian H -free graph of order at least 6 contains a connected even [2, 2k]-factor if and only if H satisfies the following condition. H≼{{K1,2k+2,Z1},{K1,k+i,Wk+3-i,0},{K1,k+i,Wk+2-i,i-1}}. And when k= 5 , 6 , we give some relevant results. We also show that for positive integers k, 2 ≤ i≤ k+ 1 and H≼ { { K_1,2k+2, Z₁} , { K_1,k+i, W_k+3-i,} } , if G is supereulerian H -free graph of order at least 6, then G contains a connected even [2, 2k]-factor. Our results extend the result of Yang et al. (Discrete Appl Math 288:192–200, 2021) and Duan et al. (Ars Comb 115:385–389, 2014).