Two Novel Evolutionary Formulations of the Graph Coloring Problem

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 distance Laplacian spectral radius and chromatic number of graphs

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

We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values χ(G) of the function chromatic number completely cover a line segment [a,b] of positive integers. Thus for an arbitrary graphical sequence d, two invariants minχ(d):=a and maxχ(d):=b naturally arise. For a regular graphical sequence d=rn:=(r,r,…,r) where r is the degree and n is the number of vertices, the exact values of a and b are found in all situations, except the case where n and r are both even and n<2r.

Maximum genus and chromatic number of graphs

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Developments in graph colouring theory were motivated by the four-colour problem and Heawood's theorem. Both of these were originally formulated as map-colouring problems that can be expressed as colouring graphs embedded on surfaces. This chapter gives an overview of the abundance of results concerning the chromatic number of graphs that are embedded on surfaces . Introduction In 1852 Francis Guthrie asked whether the regions of every planar map can be coloured with four colours in such a way that no two regions of the map with common boundary receive the same colour. In effect, by duality Guthrie was asking whether every planar graph is 4-colourable. This easily stated problem became known as the famous four-colour problem. Attempts to solve it led to many important results in graph theory, but the problem itself remained unsolved for more than a century. It was finally answered in the positive by Appel and Haken [9], [10], [12]. More about its proof comes later in this chapter. For more than a century, the four-colour problem was one of the driving forces that led to developments in graph theory, and specifically graph colouring theory. Its generalization – the Heawood problem – was the main motivation for developments in topological graph theory (see [51] or [49]). Both of these were originally formulated as map-colouring problems, asking about colouring the faces of an embedded graph so that adjacent faces receive different colours. Every map-colouring problem can be expressed as a graph-colouring problem by considering the dual graph of the map. Its vertices correspond to the faces of the map, and two vertices are adjacent if the corresponding faces are adjacent on the surface. In this chapter we discuss the colouring of graphs embedded on surfaces. We start by describing some of the ideas in the proof of the four-colour theorem and give a proof of the list-colouring version of the five-colour theorem that is due to Thomassen.

The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi-gated (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The chromatic number of a graph G, denoted χ(G) is the minimum number of colors needed to color vertices of G so that no two adjacent vertices share the same color. A functigraph over a given graph is obtained as follows: Let G' be a disjoint copy of a given G and f be a function f : V(G) → V(G'). The functigraph over G, denoted by C(G, f), is the graph with V(C(G, f)) = V(G) ∪ V(G') and E(C(G, f)) = E(G) ∪ E(G') ∪ {uv : u ∈ V(G), v ∈ V(G'), v = f(u)}. Recently, Chen et al. proved that [Formula: see text]. In this paper, we first provide sufficient conditions on functions f to reach the lower bound for any graph. We then study the attainability of the chromatic numbers of functigraphs. Finally, we extend the definition of a functigraph in different ways and then investigate the bounds of chromatic numbers of such graphs.

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.

Graph clustering via generalized colorings

On chromatic number of graphs and set-systems

The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $\deg^+(v) \le k$ or (indegree) $\deg^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, into the context of $r$-uniform hypergraphs, using a generalization of Hakimi's theorem to $r$-uniform hypergraphs and by showing some tight connections with the well known Ramsey numbers for hypergraphs.

Let D be any edge orientation of a graph G. We denote by \(\Delta _k(D)\) the maximum value t for which there exists a directed path \(v_1, \ldots , v_k\) such that \(d^{out}(v_k)=t\), where \(d^{out}(v_k)\) is the out-degree of \(v_k\) in D. We first obtain some bounds for the chromatic number of G in terms of \(\Delta _k(D)\) and then show a relationship between \(\Delta _k(D)\) and vertex partitions of a graph into degenerate subgraphs.