Let G be a nontrivial connected graph 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 subset S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbo...

An edge-coloured path is rainbow if all of its edges have distinct colours. For a connected graph G, the rainbow connection number rc(G) of G is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by a rainbow path. Similarly, the strong rainbow connection number src(G) ofG is the minimum number of colours in an edge-colouring of G such that, any tw...

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph.

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.