The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors,...

A graph G is locally irregular if adjacent vertices of G have distinct degrees. An edge colouring of G is locally irregular if each of its colours induces a locally irregular subgraph of G. The irregular chromatic index of G refers to the least number of colours used by a locally irregular edge colouring of G (if any). We propose a linear-time algorithm for determining the irregular chromatic i...

The vertex-edge Wiener index of a simple connected graph G is defined as the sum of distances between vertices and edges of G. Two possible distances D_1(u,e|G) and D_2(u,e|G) between a vertex u and an edge e of G were considered in the literature and according to them, the corresponding vertex-edge Wiener indices W_{ve_1}(G) and W_{ve_2}(G) were introduced. In this paper, we present exact form...

We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov and Zaks and presents further support for a conjecture that ∆(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analogue of a result of Robinson and the second auth...

In this paper, we study the structure of 5-critical graphs in terms of their size. In particular, we have obtained bounds for the number of major vertices in several classes of 5-critical graphs, that are stronger than the existing bounds.

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the ...

Given a simple plane graph G, an edge-face k-coloring of G is a function : E(G) [ F (G) ! f1, . . . ,kg such that, for any two adjacent or incident elements a, b 2 E(G) [ F (G), (a) 61⁄4 (b). Let e(G), ef(G), and (G) denote the edge chromatic number, the edge-face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that ef(G) 1⁄4 e(G) 1⁄4 (G) for any 2-connected...

A mixed hypergraph is characterized by the fact that it possesses C-edges as well as D-edges. In a colouring of a mixed hypergraph, every C-edge has at least two vertices of the same colour and every D-edge has at least two vertices coloured diierently. The upper and lower chromatic numbers , are the maximum and minimum numbers of colours for which there exists a colouring using all the colours...