Tabular graphs and chromatic sum

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.)

Journal of Graph Theory, 1989

The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges,

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.

Theoretical Computer Science

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χ b (G): The maximum degree ∆(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i − 1. We obtain a dichotomy result stating that for fixed k ∈ {∆(G) + 1 − p, m(G) − p}, the problem is polynomial-time solvable whenever p ∈ {0, 1} and, even when k = 3, it is NP-complete whenever p ≥ 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree ∆(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by ∆(G). Second, we show that b-Coloring is FPT parameterized by ∆(G) + k (G), where k (G) denotes the number of vertices of degree at least k.

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

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.

Lecture Notes in Computer Science, 2018

In an undirected graph, a proper (k, i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)-coloring. This is a generalization of the classic graph coloring problem. We show a parameterized algorithm for the (k, i)-coloring problem with the size of the feedback vertex set as a parameter. Our algorithm does not use tree-width machinery, thus answering a question of Majumdar, Neogi, Raman and Tale [CALDAM 2017]. We also give a faster and simpler exact algorithm for (k, k − 1)-coloring. From the hardness perspective, we show that the (k, i)coloring problem is NP-complete for any fixed values i, k, whenever i < k, thereby settling a conjecture of Méndez-Díaz and Zabala [1999] and again asked by Majumdar, Neogi, Raman and Tale. The NP-completeness result improves the partial NP-completeness shown in the preliminary version of this paper published in CALDAM 2018. ✩ This paper is the full version of the article "On the Tractability of (k, i)-Coloring"[1], published in the CALDAM 2018 conference, with the same set of authors.

2007

Let G =( V, E) be a graph. A k-coloring of a graph G is a labeling f : V (G) → T , where | T |= k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that

Given a graph = ( , ) and a set T of non-negative integers containing 0, a T -coloring of G is an integer function of the vertices of G such that | ( ) − ( )| ∉ whenever ∈ . The edge-span of a -c o l o r i n g i s t h e m a x i m u m v a l u e o f | ( ) − ( )| o v e r a l l e d g e s , a n d t h e -e d g e -s p a n of a graph G is the minimum value of the edge-span among all possible T -colorings of G. This paper discusses the T -edge span of the folded hypercube network of dimension n for the k-multiple-of-s set, = {0, , 2 , … } ∪ , where s and ≥ 1 and ⊆ { + 1, + 2, … − 1}.

Discrete Applied Mathematics, 2013

The b-chromatic number of a graph G is defined as the maximum number k of colors in a proper coloring to the vertices of G in such a way that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we show that for any connected graph G on n ≥ 5 vertices and any v ∈ Further we determine all the graphs which attain these bounds.