Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is...

Fault tolerant measures have played an important role in the reliability of an interconnection network. Edge connectivity, restricted-edge-connectivity, extra-edge-connectivity and super-edge-connectivity of many well-known interconnection networks have been explored. In this paper, we study the 2-extra-edge connectivity of a special class of graphs G(G0,G1; M) proposed by Chen et al. [Appl. Ma...

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. A graph G is internally 4-connected if it is simple, 3-connected, has at least five vertices, and if for every partition (A, B) of the edge-set of G, either |A| ≤ 3, or |B| ≤ 3, or at least four vertices of G are incident with an edge in A and an edge in B. We prove that if H and G are...

An edge colored graph G is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this work we study the rainbow connection of the random r-regular graph G = G(n, r) of order n, where r ≥ 4 is a c...

We show that a k-edge-connected graph on n vertices has at least n(k/2)n−1 spanning trees. This bound is tight if k is even and the extremal graph is the n-cycle with edge-multiplicities k/2. For k odd, however, there is a lower bound cn−1 k where ck > k/2. Specifically, c3 > 1.77 and c5 > 2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples ...

Let G = (V (G), E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such t...

Let G be a 2-edge-connected graph with o vertices of odd degree. It is well-known that one should (and can) add o 2 edges to G in order to obtain a graph which admits a nowhere-zero 2-flow. We prove that one can add to G a set of ≤ b o 4c, d2b o 5ce, and d2b o 7ce edges such that the resulting graph admits a nowhere-zero 3-flow, 4-flow, and 5-flow, respectively.

Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. We characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators.