The Topology of the Coloring Complex

In a recent paper, E. Steingrímsson associated to each simple graph G a simplicial complex _G, referred to as the coloring complex of G. Certain nonfaces of _G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial _G of G, and the reduced Euler characteristic is, up to sign, equal to |_G(–1)|–1. We show that _G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of _G, and their Euler characteristics coincide with _G(0) and –_G(1), respectively. We show for a large class of graphs—including all connected graphs—that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel.Research financed by ECs IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272.     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

An Analogue of Hajós' Theorem for the Circular Chromatic Number (II)

This paper designs a set of graph operations, and proves that for 2k/d<3, starting from K_k/_d, by repeatedly applying these operations, one can construct all graphs G with _c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with _c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003     

Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

On the Chromatic Number of the Square of the Kneser Graph <Emphasis Type="Italic">K</Emphasis>(2<Emphasis Type="Italic">k</Emphasis>+1, <Emphasis Type="Italic">k</Emphasis>)

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n–2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that (K²(2k+1, k))4k when k is odd and (K²(2k+1, k))4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on (K²(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.This work was partially supported by NSF grant DMS-0099608Final version received: April 23, 2003     

The search for chromatically unique graphs

The number of vertex-colourings of a simple graphG in not more than colours is a polynomial in. This polynomial, denoted byP(G, ), is called the chromatic polynomial ofG. A graphG is said to be chromatically unique, in short-unique, ifH G for any graphH withP(H, ) = P(G, ). Since the appearance of the first paper on-unique graphs by Chao and Whitehead in 1978, various families of and several results on such graphs have been obtained successively, especially during the last five years. It is the aim of this expository paper to give a survey on most of the works done on-unique graphs. A number of related problems and conjectures are also included.1980 Mathematical Subject Classification. Primary 05C15This work was done while the author was visiting the Department of Mathematics, National University of Singapore.     

List Colouring When The Chromatic Number Is Close To the Order Of The Graph

Ohba has conjectured [7] that if G has 2 (G)+1 or fewer vertices then the list chromatic number and chromatic number of G are equal. In this short note we prove the weaker version of the conjecture obtained by replacing 2 (G)+1 by * This research was partially supported by DIMACS and by CNRS/NSF collaboration grant. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.     

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

On the Coates–Sinnott Conjecture

Let E signify a totally real Abelian number field with a prime power conductor and ring of pintegers R _E for a prime p. Let G denote the Galois group of E over the rationals, and let be a padic character of G of order prime to p. Theorem A calculates, under a minor restriction on , the Fitting ideals of H ² _ét(R _E;Z _p (n/2+1))() over Z _p [G](). Here we require that n2 mod 4. These Fitting ideals are principal and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates–Sinnott conjecture.     

On symmetric incidence matrices of projective planes

We investigate the incidence matrix of a finite plane of ordern which admits a (C, L)-transitivityG. The elation groupG affords a generalized Hadamard matrixH=(h _ij ) of ordern and an incidence matrix for the plane is completely determined by the matrixR(H)=(R(h _ij )), whereR(g) denotes the regular permutation matrix forgG. We prove that in the caseR(H) is symmetric thatG is an elementary abelian 2-group or elseG is a nonabelian group andn is a square. Results are obtained in the abelian case linking the roots of the incidence matrixR(H) to the roots of the complex matrix (H), a nontrivial character ofG.     

An upper bound for the total chromatic number

The total chromatic number,(G), of a graphG, is defined to be the minimum number of colours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that , where(G) is the vertex chromatic number and(G) is the edge chromatic number of the graph.Partially supported by ORS grant ORS/84120     

Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture

Letg be the Lie algebra of a connected reductive groupG over an algebraically closed field of characteristicp>0. Suppose thatG ⁽¹⁾ is simply connected andp is good for the root system ofG. Ifp=2, suppose in addition thatg admits a nondegenerateG-invariant trace form. LetV be an irreducible and faithfulg-module withp-character g ^*. It is proved in the paper that dimV is divisible byp ^1/2dim() where () stands for the orbit of under the coadjoint action ofG.Oblatum 14-III-1994 & 17-XI-1994     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

Induced operators on symmetry classes of tensors

LetV be ann-dimensional inner product space,T _i ,i=1,...,k, k linear operators onV, H a subgroup ofS _m (the symmetric group of degreem), a character of degree 1 andT a linear operator onV. Denote byK(T) the induced operator ofT onV (H), the symmetry class of tensors associated withH and . This note is concerned with the structure of the setK _{, m} ^H (T₁,...,T_k) consisting of all numbers of the form traceK(T ₁ U ₁...T _k U _k ) whereU _i ,i=1,...k vary over the group of all unitary operators onV. For V=ⁿ or ⁿ, it turns out thatK _{, m} ^H (T₁,...,T_k) is convex whenm is not a multiple ofn. Form=n, there are examples which show that the convexity of _{, m} ^H (T₁,...,T_k) depends onH and .The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for his valuable advice and encouragement.     

Circular Coloring of Planar Digraphs

Let D be a digraph. The circular chromatic number ${\chi_c(D)}$ and chromatic number ${\chi(D)}$ of D were proposed recently by Bokal et?al. Let ${\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}$ . Let G be a planar graph and n?≥ 2. We prove that if the girth of G is at least ${\frac{10n-5}{3},}$ then ${\vec{\chi_c}(G)\leq \frac{n}{n-1}}$ . We also study the circular chromatic number of some special planar digraphs.     

Relaxed Locally Identifying Coloring of Graphs

A locally identifying coloring (lid-coloring) of a graph is a proper vertex-coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a relaxed locally identifying coloring (rlid-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessarily proper. We denote by \(\chi _{rlid}(G)\) the minimum number of colors used in a relaxed locally identifying coloring of a graph G. In this paper, we prove that the problem of deciding that \(\chi _{rlid}(G)=3\) for a 2-degenerate planar graph G is NP-complete and for a bipartite graph G is polynomial. We give several bounds of \(\chi _{rlid}(G)\) for different families of graphs and construct new graphs for which these bounds are tight. We also compare this parameter with the minimum number of colors used in a locally identifying coloring of a graph G (\(\chi _{lid}(G)\)), the size of a minimum identifying code of G (\(\gamma _{id}(G)\)) and the chromatic number of G (\(\chi (G)\)).     

The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n−3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group H_n−3(Δ(G)) where the dimension of the jth component in the decomposition, , equals the absolute value of the coefficient of λ^j in the chromatic polynomial of G, χ_G(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ(H), and show that the coefficient of λ^j in χ_H(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λ^j in χ_H(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.