The chromatic number of dense random graphs

The chromatic number of a graph G is defined as the minimum number of colors required for a vertex coloring where no two adjacent vertices are colored the same. The chromatic number of the dense random graph where is constant has been intensively studied since the 1970s, and a landmark result by Bollobás in 1987 first established the asymptotic value of . Despite several improvements of this result, the exact value of remains open. In this paper, new upper and lower bounds for are established. These bounds are the first ones that match each other up to a term of size o(1) in the denominator: they narrow down the coloring rate of to an explicit interval of length o(1), answering a question of Kang and McDiarmid.     

On the chromatic number of a random 5‐regular graph

It was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5‐regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5‐regular graph is asymptotically almost surely equal to 3, provided a certain four‐variable function has a unique maximum at a given point in a bounded domain. We also describe extensive numerical evidence that strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3‐colorings, where a coloring is balanced if the number of vertices of each color is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 157–191, 2009     

On the complete chromatic number of Halin graphs

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.Write.1.IntroductionDefinition1.FOrany3-connectedplanargraphG(V,E,F)withA(G)23,iftheboundaryedgesoffacefowhichisadjacenttotheothersareremoved,itbecomesatree,andthedegreeofeachvertexofV(fo)is3,andthenGiscalledaHalingraph;foiscalledtheouterfaceofG,andtheotherscalledtheinteriorfaces,thevenicesonthefacefoarecalledtheoutervenices,theoillersarecalledtheinterior...ti..,tll.ForanyplanargraphG(V,E,F),f,f'eF,fisadjacenttof'ifan…     

Circular game chromatic number of graphs

In a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ_cg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, , where is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χ_cg(G)≤2χ_a(G)(χ_a(G)+1), where χ_a(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.     

On the circular chromatic number of circular partitionable graphs

This article studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs G has . A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly 1. © 2006 Wiley Periodicals, Inc. J Graph Theory     

THE TOTAL CHROMATIC NUMBER OF PSEUDO-OUTERPLANAR GRAPHS

A Planar graph g is called a ipseudo outerplanar graph if there is a subset v.∈V(G),[V.]=i,such that G-V. is an outerplanar graph in particular when G-V.is a forest ,g is called a i-pseudo-tree .in this paper.the following results are proved;(1)the conjecture on the total coloring is true for all 1-pseudo-outerplanar graphs;(2)X1(G) 1 fo any 1-pseudo outerplanar graph g with △(G)≥3,where x4(G)is the total chromatic number of a graph g.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

Distinguishing chromatic number of random Cayley graphs

The distinguishing chromatic number of a graph $G$, denoted ${\chi }_D\left(G\right)$, is defined as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism of $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $\Gamma$ defined over certain abelian groups $A$ with $\left|A\right|=n$, and show that with probability at least $1?n^{?\Omega (logn)}$, ${\chi }_D\left(\Gamma \right)\le \chi \left(\Gamma \right)+1$.     

The total chromatic number of regular graphs of even order and high degree

The total chromatic number χ_T(G) of a graph G is the minimum number of colours needed to colour the edges and the vertices of G so that incident or adjacent elements have distinct colours. We show that if G is a regular graph of even order and , thenχ_T(G)Δ(G)+2.     

On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

Game chromatic number of outerplanar graphs

This note proves that the game chromatic number of an outerplanar graph is at most 7. This improves the previous known upper bound of the game chromatic number of outerplanar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 67–70, 1999     

On total chromatic number of planar graphs without 4-cycles

Let G be a simple graph with maximum degree A(G) and total chromatic number Xve(G). Vizing conjectured thatΔ(G) 1≤Xve(G)≤Δ(G) 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs isΔ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then Xve(G)≤8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.     

On directed local chromatic number,shift graphs,and Borsuk‐like graphs

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of “topologically t‐chromatic” graphs. We show that this minimum for large enough t‐chromatic Schrijver graphs and t‐chromatic generalized Mycielski graphs of appropriate parameters is ?t/4?+1. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 65‐82, 2010     

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G_n,p. We show that for p=p(n)n^−3/4−δ, for any fixed δ>0, the chromatic number χ(G_n,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1)log(ℓ−1)p(n−1).     

On some simple degree conditions that guarantee the upper bound on the chromatic (choice) number of random graphs

It is well known that almost every graph in the random space G(n, p) has chromatic number of order O(np/log(np)), but it is not clear how we can recognize such graphs without eventually computing the chromatic numbers, which is NP‐hard. The first goal of this article is to show that the above‐mentioned upper bound on the chromatic number can be guaranteed by simple degree conditions, which are satisfied by G(n, p) almost surely for most values of p. It turns out that the same conditions imply a similar bound for the choice number of a graph. The proof implies a polynomial coloring algorithm for the case p is not too small. Our result has several applications. It can be used to determine the right order of magnitude of the choice number of random graphs and hypergraphs. It also leads to a general bound on the choice number of locally sparse graphs and to several interesting facts about finite fields. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 201–226, 1999     

Game chromatic number of strong product graphs

The graph coloring game is a two-player game in which the two players properly color an uncolored vertex of G alternately. The first player wins the game if all vertices of G are colored, and the second wins otherwise. The game chromatic number of a graph G is the minimum integer k such that the first player has a winning strategy for the graph coloring game on G with k colors. There is a lot of literature on the game chromatic number of graph products, e.g., the Cartesian product and the lexicographic product. In this paper, we investigate the game chromatic number of the strong product of graphs, which is one of major graph products. In particular, we completely determine the game chromatic number of the strong product of a double star and a complete graph. Moreover, we estimate the game chromatic number of some King's graphs, which are the strong products of two paths.