In 1978, Chvátal and Thomassen proved that every 2-edge-connected graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value f(d) such that every 2-edge-connected graph G with diameter d has an orientation with diameter at most f(d). For d = 3, their general bounds reduce to 8 ≤ f(3) ≤ 24. We improve these bounds to 9 ≤ f(3) ≤ 11.

A graph G is Eulerian-connected if for any u and v in V (G), G has a spanning (u. v)-trail. A graph G is edge-Eulerian-connected if for any e' and elf in E(G), G has a spanning (e', elf)-trail. For an integer r ~ 0, a graph is called r-Eulerian-connected if for any X s::: E(G) with IXI ~r, and for any u, v E V(G), G has a spanning (u, v)-trail T such that X s::: E(T). The r-edge-Eulerian­ conne...

We prove that a graph admits a strongly 2-connected orientation if and only if it is 4-edge-connected, and every vertex-deleted subgraph is 2-edge-connected. In particular, every 4-connected graph has such an orientation while no cubic 3-connected graph has such an orientation.

For a connected graph G of order n ≥ 3, let f : E(G) → Zn be an edge labeling of G. The vertex labeling f ′ : V (G) → Zn induced by f is defined as f (u) = ∑ v∈N(u) f(uv), where the sum is computed in Zn. If f ′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) 6= 0 for all e ∈...

Let G = (V,E) be a connected graph. G is said to be super edge connected (or super-k for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super-k if for any edge set S # E(G) with jSj 6m, G S is still super-k. The maximum cardinality of m-super-k is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fau...

Edge-coloring total k-labeling of a connected graph G is an assignment f of non negative integers to the vertices and edges of G such that two adjacent edges e = uv and e = uv of G have different weights. The weight of an edge uv is defined by: w(e = uv) = f(u) + f(v) + f(e). In this paper, we study the chromatic number of the edge coloring by total labeling of 4-regular circulant graphs Cn(1, k).

Given two nonnegative integers s and t , a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X | ≤ s and |Y | ≤ t , there is a spanning eulerian subgraph H of G that contains X and avoids Y . We prove that if G is connected and locally k-edge-connected, thenG is (s, t)-supereulerian, for any pair of nonnegative integers s and t with s+t ≤ k−1. We further show that if s ...

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.