Supereulerianity of kk-edge-connected graphs with a restriction on small bonds

Let k≥1,l>0,m≥0k≥1,l>0,m≥0 be integers, and let Ck(l,m)Ck(l,m) denote the graph family such that a graph GG of order nn is in Ck(l,m)Ck(l,m) if and only if GG is kk-edge-connected such that for every bond S⊂E(G)S⊂E(G) with |S|≤3|S|≤3, each component of G−SG−S has order at least (n−m)/l(n−m)/l. In this paper, we show that if G∈C3(10,m)G∈C3(10,m) with n>11mn>11m, then either GG is supereulerian or it is contractible to the Petersen graph. A graph is ss-supereulerian if it has a spanning even subgraph with at most ss components. We also prove the following: if G∈C3(l,m)G∈C3(l,m) with n>(l+1)mn>(l+1)m and l≥10l≥10, then GG is ⌈(l−4)/2⌉⌈(l−4)/2⌉-supereulerian; if G∈C2(l,0)G∈C2(l,0) with 6≤l≤106≤l≤10, then GG is (l−4)(l−4)-supereulerian; if G∈C2(l,m)G∈C2(l,m) with n>(l+1)mn>(l+1)m and l≥4l≥4, then GG is (l−3)(l−3)-supereulerian.