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where the minimum is taken over all subsets X of E(G) such that ω(G − X) − c > 0. In this paper, we establish a relationship 7 between λc(G) and τc−1(G), which gives a characterization of the edge-connectivity of a graph G in terms of the spanning tree 8 packing number of subgraphs of G. The digraph analogue is also obtained. The main results are applied to(More)
In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an(More)
Let G be a graph. For any two distinct vertices x and y in G, denote distG(x, y) the distance in G from x and y. For u, v ∈ V (G) with distG(u, v) = 2, denote JG(u, v) = {w ∈ NG(u)∩NG(v)|N(w) ⊆ N [u]∪ N [v]}. A graph G is claw-free if it contains no induced subgraph isomorphic to K1,3. A graph G is called quasi-claw-free if JG(u, v) 6= ∅ for any u, v ∈ V(More)
1 A graph G is hamiltonian-connected if any two of its vertices are connected by a Hamilton 2 path (a path including every vertex of G); and G is s-hamiltonian-connected if the deletion 3 of any vertex subset with at most s vertices results in a hamiltonian-connected graph. In this 4 paper, we prove that the line graph of a (t+4)-edge-connected graph is(More)
A graph G is triangularly connected if for every pair of edges e1, e2 ∈ E(G), G has a sequence of 3-cycles C1, C2, · · · , Cl such that e1 ∈ C1, e2 ∈ Cl and such that E(Ci) ∩ E(Ci+1) 6= ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former results in(More)
Let G be a 2-edge-connected simple graph on n vertices, let A denote an abelian group with the identity element 0, and let D be an orientation of G. The boundary of a function f : E(G) → A is the function ∂ f : V (G) → A given by ∂ f (v) = ∑ e∈E+(v) f (e) − ∑ e∈E−(v) f (e), where E(v) is the set of edges with tail v and E(v) is the set of edges with head v.(More)
It is shown that every (2p+ 1) log2(|V (G)|)-edge-connected graph G has a mod (2p+ 1)orientation, and that a (4p+ 1)-regular graph G has a mod (2p+ 1)-orientation if and only if V (G) has a partition (V , V −) such that ∀U ⊆ V (G), |∂G(U)| ≥ (2p+ 1)||U ∩ V | − |U ∩ V −||. These extend former results by Da Silva and Dahad on nowhere zero 3-flows of 5-regular(More)
A graph is supereulerian if it has a spanning Eulerian subgraph. Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell ([2]) in 1997 proposed the supereulerian problem, which seeks a characterization of graphs that have spanning Eulerian subgraphs, and they indicated that this problem would be very difficult. Pulleyblank ([71]) later in 1979(More)
We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [J. of Combinatorial Theory, Ser. B. 82 (2001), 306-315] that every(More)