In this paper, we develop a polynomial time algorithm to find out all the minilnum cyclic edge cutsets of a 3-regular graph, and therefore to determine the cyclic edge connectivity of a cubic graph. The algorithm is recursive, with complexity bounded by O(n31og2 n). The algorithm shows that the number of mini~um cyclic edge cut sets of a 3-regular graph G is polynornial in v( G) and that the mi...

Global connectivity is a quite important issue for networks. The failures of some key edges may lead to breakdown of the whole system. How to find them will provide a better understanding on system robustness. Based on topological information, we propose an approach named LE (link entropy) to quantify the edge significance on maintaining global connectivity. Then we compare the LE with the othe...

The aim of edge editing or modification problems is to change a given graph by adding and deleting of a small number of edges in order to satisfy a certain property. We consider the Edge Editing to a Connected Graph of Given Degrees problem that for a given graph G, non-negative integers d, k and a function δ : V (G) → {1, . . . , d}, asks whether it is possible to obtain a connected graph G′ f...

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabă, Gregory, and Haemers. ...

Let c be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) := (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) | x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then ...

Let G be a graph. The core of G, denoted by G∆, is the subgraph of G induced by the vertices of degree ∆(G), where ∆(G) denotes the maximum degree of G. A k-edge coloring of G is a function f : E(G)→ L such that |L| = k and f(e1) 6= f(e2) for all two adjacent edges e1 and e2 of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is...

Let l > 0 and k ≥ 0 be two integers. Denote by C(l, k) the family of 2-edge-connected graphs such that a graph G ∈ C(l, k) if and only if for every bond S ⊂ E(G) with |S| ≤ 3, each component of G − S has order at least (|V (G)| − k)/l. In this paper we prove that if a 3-edge-connected graph G ∈ C(12, 1), then G is supereulerian if and only if G cannot be contracted to the Petersen graph. Our re...

Let G be a 2-edge-connected undirected graph, A be an (additive) Abelian group, and A = A − {0}. A graph G is A-connected if G has an orientation D(G) such that for every mapping b: V (G) → A satisfying  v∈V (G) b(v) = 0, there is a function f : E(G) → A ∗ such that for each vertex v ∈ V (G), the sum of f over the edges directed out from v minus the sum of f over the edges directed into v eq...

The super connectivity κ ′ and the super edge-connectivity λ′ are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ ′(L(G))= λ′(G) if and only if G is not super-λ′. As a consequence, we obtain the result of Hellwig et al. [Note on the connectiv...

Let c be a proper k-coloring of a connected graph G andΠ = (C1, C2, . . . , Ck) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c Π (v) := (d(v, C1), d(v, C2), . . . , d(v, Ck)), where d(v, Ci) = min{d(v, x)|x ∈ Ci}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c...