On the harmonious chromatic number of graphs

Discrete Mathematics, 2009

A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1, 2, . . . , k admits an adapted vertex-colouring of G by the same colours 1, 2, . . . , k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. While the ordinary chromatic number of all (categorical) powers G k of G remains the same as that of G, the adaptable chromatic number of G k may increase with k. We conjecture that for all sufficiently large k the adaptable chromatic number of G k equals the chromatic number of G. When G is complete, we prove this conjecture with k ≥ 4, and offer additional evidence suggesting it may hold with k ≥ 2. We also discuss other products and propose several open problems.

Discrete Applied Mathematics, 2005

In this paper we introduce a chromatic parameter, called the fixing chromatic number, which is related to unique colourability of graphs, in the sense that it measures how one can embed the given graph G in G ∪ K t by adding edges between G and K t to make the whole graph uniquely t-colourable. We study some basic properties of this parameter as well as its relationships to some other well-known chromatic numbers as the acyclic chromatic number. We compute the fixing chromatic number of some graph products by applying a modified version of the exponential graph construction.

Archiv der Mathematik, 1973

Theory and Applications of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors' colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ s (G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T (G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs. Proposition 1.2 (Chartrand et al.,). For n ≥ 2 and 0 ≤ t ≤ n, χ s (G n,t ) = n.

Mathematical Proceedings of the Cambridge Philosophical Society, 1968

One of the most widely studied concepts in graph theory is that of the colouring of graphs. Much of the research in this area has dealt with the chromatic number χ(G) of a graph G, defined as the minimum number of colours which can be assigned to the points of G so that adjacent points are coloured differently. One can obtain a natural generalization of this concept as follows: we define the n-chromatic number χn (G) to be the smallest number of colours which one can associate with the points of G so that not all points on any path of length n are coloured the same. The 1-chromatic number is then simply the chromatic number.

Discrete Mathematics, 1991

Miller, Z. and D. Pritikin, The harmonious coloring number of a graph, Discrete Mathematics 93 (1991) 21 l-228.

Discrete Mathematics, 2005

We determine families of circulant graphs for which each graph G = G c (n; S) has chromatic number (G) 3. In particular, we show that there exists an n 0 such that (G) 3 for all n n 0 whenever S = {s 1 , s 1 . We also prove that (G) 3 for every recursive circulant graph G = RG c (n; d), n = cd m , 1 < c d.

Discrete Mathematics, 2002

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of errorcorrecting codes. Finally, we give necessary and sufficient conditions for the injective chromatic number to be equal to the degree for a regular graph.

Journal of Discrete Mathematical Sciences and Cryptography, 2019

A k-coloring of a graph is a vertex coloring of G that uses k-colors. The chromatic number c(G) of a graph G is the minimum number of independent subsets that partition the vertex set of G. Any such minimum partition is called a chromatic partition of V(G). In this paper chromatic number of Sunlet and Bistar families of graphs are found

International Journal of Engineering & Technology, 2018

Given a simple graph , a harmonious coloring of is the proper vertex coloring such that each pair of colors seems to appears together on at most one edge. The harmonious chromatic number of , denoted by is the minimal number of colors in a harmonious coloring of . In this paper we have determined the harmonious chromatic number of some classes of Circulant Networks.