A Bound on the Total Chromatic Number

Discrete Mathematics, 2011

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number lc(G) of G is the smallest number of colors in a linear coloring of G. Let G be a graph with maximum degree ∆(G). In this paper we prove the following results: (1) lc(G) ≤ 1 2 (∆(G) 2 + ∆(G)); (2) lc(G) ≤ 8 if ∆(G) ≤ 4; (3) lc(G) ≤ 14 if ∆(G) ≤ 5; (4) lc(G) ≤ ⌊0.9∆(G)⌋ + 5 if G is planar and ∆(G) ≥ 52.

Discrete Mathematics, 1983

Vizing's Theorem, any graph G has chromatic index equal either to its maximum degree A(G) or A(G) + 1. A simple method is given for determining exactly the chromatic index of any graph with 2s + 2 vertices and maximum degree 2s.

2013

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles.

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.

1991

If G is a simple graph with minimum degree <5(G) satisfying <5(G) ^ f(| K(C?)|-f-1) the total chromatic number conjecture holds; moreover if S(G) ^ f| V(G)\ then # T (G) < A(G) + 3. Also if G has odd order and is regular with d{G) ^ \^/1\V{G)\ then a necessary and sufficient condition for ^T((7) = A((7)+ 1 is given.

1990

Vizing's Theorem, any graph G has chromatic index equal either to its maximum degree A(G) or A(G) + 1. A simple method is given for determining exactly the chromatic index of any graph with 2s + 2 vertices and maximum degree 2s.

Discrete Mathematics, 1993

Hilton, A.J.W., Recent results on the total chromatic number, Discrete Mathematics 111 (1993) 323-331.

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.

European Journal of Combinatorics, 2012

Given a graph G with maximum degree ∆, we prove that the acyclic edge chromatic number a ′ (G) of G is such that a ′ (G) ≤ 9.62∆. Moreover we prove that a ′ (G) ≤ 6.3∆ if G has girth g ≥ 5, a ′ (G) ≤ 5, 72∆ if G has girth g ≥ 7, a ′ (G) ≤ 4.52∆ if g ≥ 50 and a ′ (G) ≤ ∆ + 2 if g ≥ 26, 32∆ log ∆+ o(∆ log ∆). We further prove that the acyclic vertex chromatic number a(G) of G is such that a(G) ≤ 7, 213∆ 4/3 + o(∆ 4/3). We also prove that the star-chromatic number χ(G) of G is such that χ(G) ≤ √ 6∆ 3/2 + o(∆ 3/2). We finally prove that the β-frugal chromatic number χ β (G) of G is such that χ β (G) ≤ max{k 1 (β)∆, k 2 (β)∆ 1+1/β /(β!) 1/β }, where k 1 (β) and k 2 (β) are decreasing functions of β such that k 1 (β) ∈ [4, 6] and k 2 (β) ∈ [2, 5]. To obtain these results we use an improved version of the Lovász Local Lemma recently discovered.

Arxiv preprint arXiv:0911.4199, 2009

A dynamic coloring of a graph G is a proper coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a dynamic coloring with k colors, is called the dynamic chromatic number of G and ...