Eulerian subgraphs containing given vertices and hamiltonian line graphs

AbstractLet G be a graph and let D1(G) be the set of vertices of degree 1 in G. Veldman (1994) proves the following conjecture from Benhocine et al. (1986) that if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), d(x) + d(y) > (2n)/5 − 2, then for n large, L(G), the line graph of G, is hamiltonian. We shall show the following improvement of this theorem: if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), max[;d(x), d(y)] ⩾ n/5 − 1, then for n large, L(G) is hamiltonian with the exception of a class of well characterized graphs. Our result implies Veldman's theorem

Unique graph homomorphisms onto odd cycles, II

AbstractA natural generalization of graph colorings is graph homomorphisms. Let G and H be simple graphs. A map θ: V(G) → V(H) is called a homomorphism if θ preserves adjacency. The set of all homomorphism from G to H is denoted by Hom(G, H). A graph G is uniquely H-colorable if Hom(G, H) ≠ Π, and if for θ1, θ2 ∈ Hom(G, H), there is an automorphism π of H such that πθ1 = θ2. In this paper, we investigate some necessary necessary conditions of unique C2k+1-colorings and prove a best possible sufficient condition involving δ(G) for G to be uniquely C2k+1-colorable under some necessary conditions. This generalizes a result of Bollobás on unique C3-colorings [J. Combin. Theory Ser. B 25 (1978), 55–61]. We also find best possible conditions on the connectedness of the subgraphs of G induced by the preimages of θ, for any θ ∈ Hom(G, C2k+1)

Supereulerian graphs and the Petersen graph, II

In this note, we verify two conjectures of Catlin in [J. Graph Theory 13 (1989) 465 - 483] for graphs with at most 11 vertices. These are used to prove the following theorem which improves prior results in [10] and [13]: Let G be a 3-edge-connected simple graph with order n. If n is large and if for every edge 11.v E E(G), d(u) + d(v) 2 % - 2, then either G has a spanning eulerian subgraph or G can be contracted to the Petersen graph