On the pseudoachromatic number of join of graphs (original) (raw)
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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.
The pseudoachromatic number of a graph
2000
The pseudoachromatic number of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that, between any two distinct parts, there is at least one edge of G. This parameter is determined for graphs such as cycles, paths, wheels, certain complete multipartite graphs, and for other classes of graphs. Some open problems are raised.
Pseudoachromatic and connected-pseudoachromatic indices of the complete graph
Discrete Applied Mathematics, 2017
A complete k-coloring of a graph G is a (not necessarily proper) k-coloring of the vertices of G, such that each pair of different colors appears in an edge. A complete k-coloring is also called connected, if each color class induces a connected subgraph of G. The pseudoachromatic index of a graph G, denoted by ψ (G), is the largest k for which the line graph of G has a complete k-coloring. Analogously the connected-pseudoachromatic index of G, denoted by ψ c (G), is the largest k for which the line graph of G has a connected and complete k-coloring. In this paper we study these two parameters for the complete graph K n. Our main contribution is to improve the linear lower bound for the connected pseudoachromatic index given by Abrams and Berman [Australas J Combin 60 (2014), 314-324] and provide an upper bound. These two bounds prove that for any integer n ≥ 8 the order of ψ c (K n) is n 3/2. Related to the pseudoachromatic index we prove that for q a power of 2 and n = q 2 + q + 1, ψ (K n) is at least q 3 + 2q − 3 which improves the bound q 3 + q given by Araujo, Montellano and Strausz [J Graph Theory 66 (2011), 89-97].
On the additive chromatic number of several families of graphs
The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1,. .. , k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by Czerwiński, Grytczuk anḋ Zelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun.
On Pseudocomplete and Complete Coloring of Graphs
2013
A complete partition of a graph G = (V, E) is a partition of the vertex set V in which there is an edge connecting every pair of distinct classes. That is, a partition V 1 ,…,V t of V is complete if, for every i, j, i ≠ j, there is an edge {v i , v j } such that v i ∈ V i and v j ∈ V j. In this paper we study the problem of finding a complete partition of the largest possible size for certain finite, simple and undirected graphs. This concept is interpreted in terms of graph vertex colorings. In the process we give (1)illustrations, (2)prove results, (3)indicate the scope of its application and also raise some open problems.
On the edge coloring of graph products
International Journal of Mathematics and Mathematical Sciences, 2005
The edge chromatic number ofGis the minimum number of colors required to color the edges ofGin such a way that no two adjacent edges have the same color. We will determine a sufficient condition for a various graph products to be of class 1, namely, strong product, semistrong product, and special product.
On the pseudoachromatic index of the complete graph
Journal of Graph Theory, 2010
Let q = 2 β be, for some β ∈ IN, and let n = q 2 + q + 1. By exhibiting a complete colouring of the edges of K n , we show that the pseudoachromatic number of the complete line graph G n = L(K n) is at least ψ(G n) ≥ q 3 + q-this bound improves the implicit bound of Jamison [9] which is given in terms of the achromatic number: ψ(G n) ≥ α(G n) ≥ q 3 + 1. We also calculate, precisely, the pseudoachromatic number when q + 1 extra points are added: ψ(G n+q+1) = q 3 + 2q 2 + 3q.
On the pseudoachromatic index of the complete graph II
Boletin De La Sociedad Matematica Mexicana, 2014
Let ψ1(Kn) and α1(Kn) be the pseudoachromatic index and the achromatic index of the complete graph respectively. Let γ ≥ 2 be a positive integer q = 2 γ and m = (q + 1) 2. In this paper exhibiting three closely related edge-colorings of the complete graph we prove that for a ∈ {0, 1, 2}: ψ1(Km−a) = α1(Km−a) = q(m − 2a).
On the Pseudoachromatic Index of the Complete Graph III
Graphs and Combinatorics, 2018
Let Π q be the projective plane of order q, let ψ(m) := ψ(L(K m)) the pseudoachromatic number of the complete line graph of order m, let a ∈ {3, 4,. .. , q 2 + 1} and m a = (q + 1) 2 − a. In this paper, we improve the upper bound of ψ(m) given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89-97] and Jamison [Discrete Math. 74 (1989), 99-115] in the following values: if x ≥ 2 is an integer and m ∈ {4x 2 − x,. .. , 4x 2 + 3x − 3} then ψ(m) ≤ 2x(m − x − 1). On the other hand, if q is even and there exists Π q we give a complete edge-colouring of K ma with (m a − a)q colours. Moreover, using this colouring we extend the previous results for a =