Coloring sums of extensions of certain graphs (original) (raw)
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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.
Generalised colouring sums of graphs
Cogent Mathematics, 2016
The notion of the b-chromatic number of a graph attracted much research interests and recently a new concept, namely the b-chromatic sum of a graph, denoted by � (G), has also been introduced. Motivated by the studies on b-chromatic sum of graphs, in this paper we introduce certain new parameters such as-chromatic sum, +-chromatic sum, b +-chromatic sum,-chromatic sum and +-chromatic sum of graphs. We also discuss certain results on these parameters for a selection of standard graphs.
Certain Chromatic Sums of Some Cycle Related Graph Classes
Discrete Mathematics, Algorithms and Applications, 2016
Let [Formula: see text] be a certain type of proper [Formula: see text]-coloring of a given graph [Formula: see text] and [Formula: see text] denote the number of times a particular color [Formula: see text] is assigned to the vertices of [Formula: see text]. Then, the coloring sum of a given graph [Formula: see text] with respect to the coloring [Formula: see text], denoted by [Formula: see text] is defined to be [Formula: see text]. The coloring sums such as [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, etc. are some of these types of coloring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle-related graphs.
On New Thue Colouring Concepts of Certain Graphs
arXiv: Combinatorics, 2016
The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in GGG is non-repetitive. The change in the Thue number, pi(G)\pi(G)pi(G), as edges are iteratively removed from a graph GGG is studied. The notion of the tau\tautau-index denoted, tau(G)\tau(G)tau(G), of a graph GGG is introduced as well. tau(G)\tau(G)tau(G) serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.
The chromatic sum Sigma(G)\Sigma(G)Sigma(G) of a graph GGG is the smallest sum of colors among of proper coloring with the natural number. We present an upper bound for the chromatic sum of GGG that it is relevant to the existence of homomorphism from GGG to Kneser Graph KG(m,n)KG(m,n)KG(m,n). Also we introduce a lower bound for the chromatic sum of vertex transitive graphs.
On the Theory of Colorful Graphs
2007
The theory of colorful graphs can be developed by working in Galois field modulo (p), p > 2 and a prime number. The paper proposes a program of possible conversion of graph theory into a pleasant colorful appearance. We propose to paint the usual black (indicating presence of an edge) and white (indicating absence of an edge) edges of graphs using multitude of colors and study their properties. All colorful graphs considered here are simple, i.e. not having any multiple edges or self-loops. This paper is an invitation to the program of generalizing usual graph theory in this direction.
Topics in Chromatic Graph Theory
2009
Chromatic graph theory is a thriving area that uses various ideas of 'colouring' (of vertices, edges, etc.) to explore aspects of graph theory. It has links with other areas of mathematics, including topology, algebra and geometry, and is increasingly used in such areas as computer networks, where colouring algorithms form an important feature. While other books cover portions of the material, no other title has such a wide scope as this one, in which acknowledged international experts in the field provide a broad survey of the subject. All 15 chapters have been carefully edited, with uniform notation and terminology applied throughout. Bjarne Toft (Odense, Denmark), widely recognized for his substantial contributions to the area, acted as academic consultant. The book serves as a valuable reference for researchers and graduate students in graph theory and combinatorics and as a useful introduction to the topic for mathematicians in related fields.
General and acyclic sum-list-colouring of graphs
Applicable Analysis and Discrete Mathematics, 2016
We investigate list colouring of a graph in which the sizes of the lists assigned to different vertices can be different. For a given graph G and a class of graphs P we colour G from the lists in such a way that each colour class induces a graph in P. The aim is to find the P-sum-choice-number of G, which means the smallest possible sum of all the list sizes such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes. We prove several general results concerning the P-sum-choice-number of an arbitrary graph. Using some of them, we also estimate or, in the case of complete graphs or some complete bipartite graphs, exactly determine the P-sum-choice-number of a graph, when P is the class of acyclic graphs.
Total Thue colourings of graphs
A total colouring of a graph is a colouring of its vertices and edges such that no two adjacent vertices or edges have the same colour and moreover, no edge coloured c has its endvertex coloured c too. A weak total Thue colouring of a graph G is a colouring of its vertices and edges such that the colour sequence of consecutive vertices and edges of every path of G is nonrepetitive. In a total Thue colouring also the induced vertex-colouring and edge-colouring of G are nonrepetitive. The weak total Thue number πT w (G) of a graph G denotes the minimum number of colours required in every weak total Thue colouring and the minimum number of colours required in every total Thue colouring is called the total Thue number πT. Here we show some upper bounds for both parameters depending on the maximum degree or size of the graph. We also give some lower bounds and some better upper bounds for these graph parameters considering special families of graphs.