On chromatic number of graphs and set-systems

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

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

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

We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The first formulation is based on the relationship that exists between a graph's chromatic number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The second formulation, unlike the first one, does not tackle one graph at a time, but rather aims at evolving a “program” to color all graphs belonging to a class whose members all have the same number of nodes and other common attributes. The heuristics that result from these formulations have been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and have been found to be competitive when compared to the several other heuristics that have also been tested on those graphs.

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.

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)

Graph clustering via generalized colorings

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.

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdős [8] conjectured that |C(G)|=Ω ` d (g−1)/2 for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length Ω ` d (g−1)/2 . The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle [7,11,21,24,25]. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that Ω ` d (g−1)/2 is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d.

The Steiner <span class="math"><em>n</em></span>-antipodal graph of a graph <span class="math"><em>G</em></span> on <span class="math"><em>p</em></span> vertices, denoted by <span class="math"><em>S</em><em>A</em>_<em>n</em>(<em>G</em>), </span> has the same vertex set as <span class="math"><em>G</em></span> and any <span class="math"><em>n</em>(2 ≤ <em>n</em> ≤ <em>p</em>)</span> vertices are mutually adjacent in <span class="math"><em>S</em><em>A</em>_<em>n</em>(<em>G</em>)</span> if and only if they are <span class="math"><em>n</em></span>-antipodal in <span class="math"><em>G</em>.</span> When <span class="math"><em>G</em></span> is disconnected, any <span class="math"><em>n</em></span> vertices are mutually adjacent in <span class="math"><em>S</em><em>A</em>_<em>n</em>(<em>G</em>)</span> if not all of them are in the same component. <span class="math"><em>S</em><em>A</em>_<em>n</em>(<em>G</em>)</span> coincides with the antipodal graph <span class="math"><em>A</em>(<em>G</em>)</span> when <span class="math"><em>n</em> = 2</span>. The least positive integer <span class="math"><em>n</em></span> such that <span class="math"><em>S</em><em>A</em>_<em>n</em>(<em>G</em>) ≅ <em>H</em></span>, for a pair of graphs <span class="math"><em>G</em></span> and <span class="math"><em>H</em></span> on <span class="math"><em>p</em></span> vertices, is called the Steiner <span class="math"><em>A</em></span>-completion number of <span class="math"><em>G</em></span> over <span class="math"><em>H</em>.</span> When <span class="math"><em>H</em> = <em>K</em>_<em>p</em>, </span> the Steiner <span class="math"><em>A</em></span>-completion number of <span class="math"><em>G</em></span> over <span class="math"><em>H</em></span> is called the Steiner antipodal number of <span class="math"><em>G</em>.</span> In this article, we obtain the Steiner antipodal number of some families of graphs and for any tree. For every positive integer <span class="math"><em>k</em>, </span> there exists a tree having Steiner antipodal number <span class="math"><em>k</em></span> and there exists a unicyclic graph having Steiner antipodal number <span class="math"><em>k</em></span>. Also we show that the notion of the Steiner antipodal number of graphs is independent of the Steiner radial number, the domination number and the chromatic number of graphs.