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In previous papers, Catlin introduced four functions, denoted S O , S R , S C , and S H , between sets of ÿnite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-ows for a ÿxed integer k ¿ 3. Unfortunately, a subtle error caused several… (More)

In this paper we prove that if k is an integer no less than 3, and if G is a two-connected graph with 2n-a vertices, a E {0, 1}, which is regular of degree n-k, then G is Hamiltonian if a = 0 and n > k 2 + k + 1 or if a = I and n > 2k 2-3k =, 3. We use the notation and terminology of [1]. Gordon [4] has proved that there are only a small number of… (More)

In this paper, we show that if n 14 and if G is a Z-connected graph with 2n or 2n-1 vertices which is regular of degree n-2, then G is Hamiltonian if and only if G is not the Petersen graph. We use the terminology of Behzad and Chartrand [2]. In particular, a set of vertices in a graph is independent if no two of the vertices in the set are adjacent. A… (More)