Chromatic Number Of Graph Research Articles

Total chromatic number of graphs with small genus

Abstract Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.

On the Chromatic Number of Graphs

Computing the chromatic number of a graph is an NP-hard problem. For random graphs and some other classes of graphs, estimators of the expected chromatic number have been well studied. In this paper, a new 0–1 integer programming formulation for the graph coloring problem is presented. The proposed new formulation is used to develop a method that generates graphs of known chromatic number by using the KKT optimality conditions of a related continuous nonlinear program.

Star chromatic numbers of some planar graphs

The concept of the star chromatic number of a graph was introduced by Vince (A. Vince, Star chromatic number, J. Graph Theory 12 (1988), 551–559), which is a natural generalization of the chromatic number of a graph. This paper calculates the star chromatic numbers of three infinite families of planar graphs. More precisely, the first family of planar graphs has star chromatic numbers consisting of two alternating infinite decreasing sequences between 3 and 4; the second family of planar graphs has star chromatic numbers forming an infinite decreasing sequence between 3 and 4; and the third family of planar graphs has star chromatic number 7/2. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 33–42, 1998

On some conjectures of Graffiti

The computer program Galatea Gabriella Graffiti [8] made several conjectures concerning the chromatic numbers of graphs, related to the work of DeLaVina and Fajtlowicz [5]. We prove one of these conjectures, and disprove four.

Orthogonal representations over finite fields and the chromatic number of graphs

We study the relationship between the minimum dimension of an orthogonal representation of a graph over a finite field and the chromatic number of its complement. It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not. This implies the NP-hardness of determining the minimum rank of a matrix in such a class. Finally we give for any class of matrices defined by a graph that is interesting in this respect a reduction of the 3-colorability problem to the problem of deciding whether or not this class contains a matrix of rank equal to three.

Ramseyan Properties of Graphs

Every graph of chromatic number $k$ with more than $k(r-1)(b-1)$ vertices has a $b$-element independent set of vertices such that if any two of them are joined by an edge then the chromatic number stays the same or a $r$-element independent set of vertices such that joining any two of them by an edge increases the chromatic number.

Applications of hypergraph coloring to coloring graphs not inducing certain trees

We present a simple result on coloring hypergraphs and use it to obtain bounds on the chromatic number of graphs which do not induce certain trees.

Total Chromatic Number of Graphs of Order 2n + l having Maximum Degree 2n − 1

Total Chromatic Number of Graphs of Order 2<i>n</i> + l having Maximum Degree 2<i>n</i> − 1

Relaxed chromatic numbers of graphs

Given any family of graphsP, theP chromatic number ? p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.

Generalized Chromatic Numbers of Random Graphs

Given a hereditary (closed under taking induced subgraphs) class of graphs $\mathcal{P}$, the $\mathcal{P}$-chromatic number of a graph G, denoted $\chi _\mathcal{P} ( G )$, is defined to be the least integer k such that $V( G )$ admits a partition into k subsets each of which induce a member of $\mathcal{P}$. The $\mathcal{P}$-chromatic number of random graphs G on n vertices with fixed edge probability $0 < p < 1$ is studied and it is shown that $\chi _\mathcal{P} ( G ) \sim cn$ in case $\mathcal{P} < \infty $ and $\chi _\mathcal{P} ( G ) = \Theta ( n/\log n )$ in case $| \mathcal{P} | = \infty $. Also considered are generalized edge chromatic numbers.

?-Perfect graphs

ω-Perfect graph is defined and some classes of ω-perfect graphs are described, although the characterization of the complete class of ω-perfect graphs remains an open question. A bound on the chromatic number for graphs without even holes is derived.

The chromatic number of graphs which induce neither K1,3 nor K5− e

If G is a graph which induces neither K 1,3 nor K 5− e and if Δ( G)⩽2 ω( G)−5, then χ( G) = ω( G). Conversely, for each k ⩾ 4 there is a graph G which induces neither K 1,3 nor K 5− e such that ω( G) = k, Δ( G) = 2 k − 3 and χ( G) = k + 1.

Graphes cubiques d'indice trois, graphes cubiques isochromatiques, graphes cubiques d'indice quatre

The process introduced by E. Johnson [ Amer. Math. Monthly 73 (1966), 52–55] for constructing connected cubic graphs can be modified so as to obtain restricted classes of cubic graphs, in particular, those defined by their chromatic number or their chromatic index. We construct here the graphs of chromatic number three and the graphs whose chromatic number is equal to its chromatic index (isochromatic graphs). We then give results about the construction of the class of graphs of chromatic index four, and in particular, we construct an infinite class of “snarks.”

A bound on the chromatic number of graphs without certain induced subgraphs

For each n, we obtain a (polynomial) bound on the chromatic number in terms of the maximum size of a complete subgraph, for graphs not having n · K 2 as an induced subgraph.

On an upper bound of a graph's chromatic number, depending on the graph's degree and density

Grünbaum's conjecture on the existence of k-chromatic graphs of degree k and girth g for every k ≥ 3, g ≥ 3 is disproved. In particular, the bound obtained states that the chromatic number of a triangle-free graph does not exceed [ 3(σ + 2) 4 ], where σ is the graph's degree.

A generalization of a theorem of de Bruijn and Erdös on the chromatic number of infinite graphs

The de Bruijn-Erdös theorem states that the chromatic number of a graph is n (a finite number) if and only if the chromatic numbers of all its finite subgraphs are ≤ n. We prove a generalization of this result.

Chromatic numbers of subgraphs

Given an arbitrary graph, wh~t can we say about the setl of all chromatic numbers of its subgraphs ? In particular, for e~rdinals m ~ n, does every graph of chromatic number m have a subgraph of chromatic number n? For n ~ ~t 0 the affirmative answer follows from the theorem of DE BI~UIJ~ and ERD6S [1], SO the simplest nontrivial question is whether every graph of chromatic number ~2 has a subgraph of chromatic number Nr I have no positive results; the purpose of this note is only to state the probleml in the form of 2 dubious conjectures, and to prove that the stronger conjecture implies a weak form of the GCH. If % is a set, [S] 2 = {X c~ S " [X l = 2}. We regard a graph as a system G = (V, E) consisting of a set V of vertices and a set E ~c [ V] ~ of edges. A coloring of G is a function f defined on V such thatf(x) :~f(y) whenever {x, y} # E. G is m-colorable if there is a coloring f such that If(V) l ~ m. The chromatic number of G, ehr(G), is the least m such that G is m-eolorable. A graph G" ---- (V', E') is a vertex subgraph of G if V' ~ V and E' = E N [V'~]; an edge subgraph if V' ~ V and E' c E. ConJEcturE 1. For m ~ n, every graph of chromatic number m has a vertex subgraph of chromatic number n.

On A Combinatorial Problem III

A family of sets {Aα} is said by Miller [3] to have property B if there exists a set S which meets all the sets Aα and contains none of them. Property B has been extensively studied in several recent papers (see the references in [2] and the last chapter of P. Erdös and A. Hajnal, On chromatic number of graphs and set systems, Acta. Math. Acad. Sci. Hung. 17 (1966) 61–99).

On chromatic number of graphs and set-systems

On chromatic number of graphs and set-systems

Total Digraphs

The line - graph of an ordinary graph G is that graph whose points can be put in one-to-one correspondence with the lines of G in such a way that two points of are adjacent if and only if the corresponding lines of G are adjacent. This concept originated with Whitney [ 5 ], has the property that its (point) chromatic number equals the line chromatic number of G, where the point (line) chromatic number of graph is the minimum number of colors required to color the points (lines) of the graph such that adjacent points (lines) are colored differently.