On the harmonious chromatic number of a graph

European Journal of Combinatorics, 2008

The adaptable chromatic number of a graph G is the smallest integer k such that for any edge k-colouring of G there exists a vertex kcolouring of G in which the same colour never appears on an edge and both its endpoints. (Neither the edge nor the vertex colourings are necessarily proper in the usual sense.)

Discrete Mathematics

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χ pcf (G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χ pcf (G) in terms of other graph parameters. In particular, we show that χ pcf (G) ≤ 5∆(G) 2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4 ≤ k ≤ 6. The paper concludes with few open problems.

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 Applied Mathematics, 1999

is the maximum k such that V has a partition VI, Vz, , V, into independent sets, the union of no pair of which is independent.

The concept of vertex coloring pose a number of challenging open problems in graph theory. Among several interesting parameters, the coloring parameter, namely the pseudoachromatic number of a graph stands a class apart. Although not studied very widely like other parameters in the graph coloring literature, it has started gaining prominence in recent years. The pseudoachromatic number of a simple graph G, denoted ψ(G), is the maximum number of colors used in a vertex coloring of G, where the adjacent vertices may or may not receive the same color but any two distinct pair of colors are represented by at least one edge in it. In this paper we have computed this parameter for a number of classes of graphs.

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.

Graphs and Combinatorics, 1990

The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that)('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and)((G)is the edge chromatic number of the graph.

Indian Journal of Pure and Applied Mathematics

We introduce the equitable distinguishing chromatic number v ED ðGÞ of a graph G as the least number k such that G has an equitable coloring with k colors that is only preserved by the trivial automorphism. The equitable distinguishing chromatic number of some graphs such as paths, cycles, trees, and Cartesian products of two complete graphs are determined. Keywords Distinguishing chromatic number Á Graph automorphism Á Equitable coloring Á Equitable distinguishing chromatic number Mathematics Subject Classification 05C15 Á 05C25 1 Introduction Albertson and Collins [1] introduced the distinguishing number of a graph. Let G be an undirected simple graph and let r be a positive integer. A coloring h : VðGÞ ! f1;. . .; rg of the vertices of G is said to be rdistinguishing provided that no nontrivial automorphism of G preserves all of the vertex color. The distinguishing number of G, denoted by D(G), is the smallest integer r such that G has an r-distinguishing coloring. Collins and Trenk [10] studied some properties of distinguishing chromatic number of graphs. The distinguishing chromatic number of a graph G, denoted by v D ðGÞ, is the minimum r such that G has a proper r-distinguishing coloring. For more results in this direction, the interested reader may like to consult [4-7, 9, 15, 18]. On the other hand, the notion of the equitable colorability was introduced by Meyer [21]. A graph G ¼ ðV; EÞ is said to be equitably k-colorable, where k is a positive integer, if V may be partitioned into independent sets V 1 ;. . .; V k such that jjV i j À jV j jj 1 for any i and j with 1 i; j k and i 6 ¼ j. The equitable chromatic number of G, denoted by v e ðGÞ, is defined as the smallest k such that G is equitably k-123

Discrete Mathematics, 1989

Tke s&chromatic number X,(G) of a graph G = (V, E) is the smallest order k of a partition {v,, v,,-* * 9 V,} of the vertices V(G) such that the subgraph (V;:) induced by each subset V;: consists of a disjoint union of complete subgraphs. By definition, Xs(G) <X(G), the chromatic number of G. This paper develops properties of this lower bound for the chromatic number. 1. Introduction An n-coloring of a graph G = (V, E) is a function f from V onto N = 11 2 l l 9 n} such that whenever vertices u and v are adjacent, thenf(u) #f(v). Ai e'q;ivalent definition, which provides much of the motivation for this paper, is that an n-coloring of G is a partition {VI, V2,. .. , Vn} of the vertices V(G) into color classes, such that for every i = 1,2,. .. , n, the subgraph (6) induced by Vj is totally disconnected, or equivalently, is the disjoint union of K,'s (complete graphs with one vertex). A partition {V', V2,. .. , Vn} of V(G) is called complete ifforeveryi,j, lsi<lm < n, there exists a vertex u in & and a vertex u in I$ such that u and v are adjacent. The chromatic number X(G) and the achromatic number Y(G) are the smallest and largest integers n, respectively, for which G has a complete n-coloring. Notice that in the definition of the chromatic number the completeness of the partition is not required but follows easily from the definition, whereas the completeness of the partition used in defining the achromatic number is essential. The chromatic number, of course, is a very well studied parameter, whose his&y dates back to the famous Four Color Problem and the early work of Kempe in 1879 [25] and Heawood in 1890 [22]. The achromatic number was first studied as a parameter by Harary, Hedetniemi and Prins in 1967 [20], and later named and studied by Harary and Hedetniemi in 1970 [ 191. This paper was motivated in part by an interest in developing interesting lower d Research supported in part by Office of Naval Research Contract NOOO14-86-K-0693.

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.