Khee Meng Koh

The search for chromatically unique graphs

The number of vertex-colourings of a simple graphG in not more thanλ colours is a polynomial inλ. This polynomial, denoted byP(G, λ), is called the chromatic polynomial ofG. A graphG is said to be chromatically unique, in shortχ-unique, ifH ≅ G for any graphH withP(H, λ) = P(G, λ). Since the appearance of the first paper onχ-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 onχ-unique graphs. A number of related problems and conjectures are also included.

Kings in multipartite tournaments

Abstract Let T be an n -partite tournament and let k r ( T ) denote the number of r -kings of T . Gutin (1986) and Petrovic and Thomassen (1991) proved independently that if T contains at most one transmitter, then k 4 ( T ) ⩾ 1, and found infinitely many bipartite tournaments T with at most one transmitter such that k 3 ( T ) = 0. In this paper, we (i) obtain some sufficient conditions for T to have k 3 ( T ) ⩾ 1, (ii) show that if T contains no transmitter, then k 4 ( T ) ⩾ 4 when n = 2, and k 4 ( T ) ⩾ 3 when n ⩾ 3, and (iii) characterize all T with no transmitter such that the equalities in (ii) hold.

On optimal orientations of Cartesian products of even cycles

For a graph G, let D(G) be the family of strong orientations of G. Define a(G) = min {d(D) D E D(G)} and ρ(G) = a(G) - d(G), where d(D) [respectively, d(G)] denotes the diameter of the digraph D (respectively, graph G). Let G × H denote the Cartesian product of the graphs G and H, and C p , the cycle of order p. In this paper, we show that ρ(C 2m × C 2n ) = 0 and ρ(C 2m × C 2n × G 1 × G 2 ×… × G k ) = 0, where {G i | 1 ≤ i ≤ k} is any combination of paths and cycles.

On Graphs Having No Chromatic Zeros in (1,2)

For a graph

The orientation number of two complete graphs with linkages

G

Bounds for the real zeros of chromatic polynomials

of order

On Zero-Free Intervals in

nge 2

(1,2)

, an ordering

of Chromatic Polynomials of Some Families of Graphs

(x_1,x_2,ldots, x_n)

An Upper Bound for the Total Restrained Domination Number of Graphs

of the vertices in