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

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