Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

On lower bounds for the chromatic number in terms of vertex degree

In this paper we discuss the existence of lower bounds for the chromatic number of graphs in terms of the average degree or the coloring number of graphs. We obtain a lower bound for the chromatic number of K_1,t-free graphs in terms of the maximum degree and show that the bound is tight. For any tree T, we obtain a lower bound for the chromatic number of any K_2,t-free and T-free graph in terms of its average degree. This answers affirmatively a modified version of Problem 4.3 in [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995]. More generally, we discuss δ-bounded families of graphs and then we obtain a necessary and sufficient condition for a family of graphs to be a δ-bounded family in terms of its induced bipartite Turán number. Our last bound is in terms of forbidden induced even cycles in graphs; it extends a result in [S.E. Markossian, G.S. Gasparian, B.A. Reed, β-perfect graphs, J. Combin. Theory Ser. B 67 (1996) 1–11].     

On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a directed cycle on 5 vertices which have oriented chromatic number 5. This bound is tight for k?4.     

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     

Colourings of oriented connected cubic graphs

In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the oriented chromatic number for the family of orientations of connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena (Sopena, 1997) regarding the chromatic number of orientations of connected cubic graphs is false.     

Bounds for the b-chromatic number of some families of graphs

In this paper we obtain some upper bounds for the b-chromatic number of K_1,s-free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are given in terms of either the clique number or the chromatic number of a graph or the biclique number for a bipartite graph. We show that all the bounds are tight.     

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

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. * Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826, and NK62321. † Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046234, AT048826, and NK62321.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

Homomorphism bounds for oriented planar graphs

If is a class of oriented graphs (directed graphs without opposite arcs), then an oriented graph is a homomorphism bound for if there is a homomorphism from each graph in to H. We find some necessary conditions for a graph to be a homomorphism bound for the class of oriented planar graphs and prove that such a graph must have maximum degree at least 16; thus there exists an oriented planar graph with oriented chromatic number at least 17. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 175–190, 2007     

Bounds on the localization number

We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written as ζ(G). We settle a conjecture of Bosek et al by providing an upper bound on the chromatic number as a function of the localization number. In particular, we show that every graph with ζ(G) ≤ k has degeneracy less than 3^k and, consequently, satisfies χ(G) ≤ 3^ζ(G). We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.     

On extremal sizes of locally k-tree graphs

A graph G is a locally k-tree graph if for any vertex v the subgraph induced by the neighbours of v is a k-tree, k ⩾ 0, where 0-tree is an edgeless graph, 1-tree is a tree. We characterize the minimum-size locally k-trees with n vertices. The minimum-size connected locally k-trees are simply (k + 1)-trees. For k ⩾ 1, we construct locally k-trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an n-vertex locally k-tree graph is between Ω(n) and O(n ²), where both bounds are asymptotically tight. In contrast, the number of edges in an n-vertex k-tree is always linear in n.     

A study on oriented relative clique number

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset $X$ of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least $g$ and planar graphs with girth at least $g+2$ for all $g\ge 3$. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).     

New upper bounds on harmonious colorings

We present an improved upper bound on the harmonious chromatic number of an arbitrary graph. We also consider ?fragmentable”? classes of graphs (an example is the class of planar graphs) that are, roughly speaking, graphs that can be decomposed into bounded-sized components by removing a small proportion of the vertices. We show that for such graphs of bounded degree the harmonious chromatic number is close to the lower bound (2m)^1/2, where m is the number of edges.     

定向图的斜秩

定向图Gσ是一个不含有环(loop)和重边的有向图,其中G称作它的基图.S(Gσ)是Gσ的斜邻接矩阵.S(Gσ)的秩称为Gσ的斜秩,记为sr(Gσ).定向图的斜邻接矩阵是斜对称的,因而,它的斜秩是偶数.本文主要考虑简单定向图的斜秩,首先给出斜秩的一些简单基本知识,紧接着分别刻画斜秩是2的定向图和斜秩是4的带有悬挂点的定向图;其次利用匹配数给出具有n个顶点、围长是k的单圈图的斜秩表达式;作为推论,列出斜秩是4的所有单圈图和带有悬挂点的双圈图;另外研究具有n个顶点、围长是k的单圈图的图类中斜秩的最小值,并刻画了极图;最后研究斜邻接矩阵是非奇异的定向单圈图.     

关于图的3-条件色数

For integers k0,r0,a(k,r)-coloring of a graph G is a proper k-coloring of the vertices such that every vertex of degree d is adjacent to vertices with at least min{d,r}diferent colors.The r-hued chromatic number,denoted byχr(G),is the smallest integer k for which a graph G has a(k,r)-coloring.Define a graph G is r-normal,ifχr(G)=χ(G).In this paper,we present two sufcient conditions for a graph to be 3-normal,and the best upper bound of 3-hued chromatic number of a certain families of graphs.     

Bounds on Special Subsets in Graphs, Eigenvalues and Association Schemes

We give a bound on the sizes of two sets of vertices at a given minimum distance in a graph in terms of polynomials and the Laplace spectrum of the graph. We obtain explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance. For graphs with four eigenvalues we find bounds on the number of vertices that are not adjacent to a given vertex and that have µ common neighbours with that vertex. Furthermore we find that the regular graphs for which the bounds are tight come from association schemes.     

On the oriented chromatic index of oriented graphs

A homomorphism from an oriented graph G to an oriented graph H is a mapping from the set of vertices of G to the set of vertices of H such that is an arc in H whenever is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (the line digraph LD(G) of G is given by V(LD(G)) = A(G) and whenever and ). We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree. We also consider lower and upper bounds of oriented chromatic number in terms of oriented chromatic index. We finally prove that the problem of deciding whether an oriented graph has oriented chromatic index at most k is polynomial time solvable if k ≤ 3 and is NP‐complete if k ≥ 4. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 313–332, 2008     

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic numberχ^k(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate χ^k for unit disk graphs and random unit disk graphs to generalise results of McDiarmid and Reed [Colouring proximity graphs in the plane, Discrete Math. 199(1-3) (1999) 123-137], McDiarmid [Random channel assignment in the plane, Random Structures Algorithms 22(2) (2003) 187-212], and McDiarmid and Müller [On the chromatic number of random geometric graphs, submitted for publication].     

Acyclic and oriented chromatic numbers of graphs

The oriented chromatic number χ_o(G ^→) of an oriented graph G ^→ = (V, A) is the minimum number of vertices in an oriented graph H ^→ for which there exists a homomorphism of G ^→ to H ^→. The oriented chromatic number χ_o(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for χ_o(G) in terms of χ_a(G). An upper bound for χ_o(G) in terms of χ_a(G) was given by Raspaud and Sopena. We also give an upper bound for χ_o(G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. © 1997 John Wiley & Sons, Inc.     

Coloring the square of a planar graph

We prove that for any planar graph G with maximum degree Δ, it holds that the chromatic number of the square of G satisfies χ(G²) ≤ 2Δ + 25. We generalize this result to integer labelings of planar graphs involving constraints on distances one and two in the graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 110–124, 2003