Kee L. Teo

The search for chromatically unique graphs—II

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.

Non-chordal graphs having integral-root chromatic polynomials II

It is known that the chromatic polynomial of any chordal graph has only integer roots. However, there also exist non-chordal graphs whose chromatic polynomials have only integer roots. Dmitriev asked in 1980 if for any integer p ≥ 4, there exists a graph with chordless cycles of length p whose chromatic polynomial has only integer roots. This question has been given positive answers by Dong and Koh for p = 4 and p = 5. In this paper, we construct a family of graphs in which all chordless cycles are of length p for any integer p ≥ 4. It is shown that the chromatic polynomial of such a graph has only integer roots iff a polynomial of degree p - 1 has only integer roots. By this result, this paper extends Dong and Kohs result for p = 5 and answer the question affirmatively for p = 6 and 7.

Chromaticity of some families of dense graphs

For a graph G, let P(G; � ) be its chromatic polynomial and let [G] be the set of graphs having P(G; � ) as their chromatic polynomial. We call [G] the chromatic equivalence class of G .I f [G ]= {G}, then G is said to be chromatically unique. In this paper, we 4rst determine [G] for each graph G whose complement 5 G is of the form aK1 ∪bK3 ∪ � 16i6s Pli , where a; b are any nonnegative integers and li is even. By this result, we 4nd that such a graph G is chromatically unique i7 ab = 0 and li � 4 for all i. This settles the conjecture that the complement of Pn is chromatically unique for each even n with n � 4. We also determine [H ] for each graph H whose complement 5 H is of the form aK3 ∪ � 16i6s Pui ∪ � 16j6t Cvj , where ui ? 3 and ui � 4 (mod 5) for all i. We prove that such a graph H is chromatically unique if ui +1 � vj for all i; j and ui is even when ui ? 6. c � 2002 Elsevier Science B.V. All rights reserved.

Chromatic classes of 2-connected ( n,n +3)-graphs with at least two triangles

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ~ H, if P(G) = P(H). A graph G is chromatically unique if G ? H for any graph H such that H ~ G. Let H denote the class of 2-connected graphs with n vertices and n + 3 edges which contain at least two triangles. It follows that if G ? H and H ~ G, then H ? H. In this paper, we determine all equivalence classes in H under the equivalence relation ~ and characterize the structure of the graphs in each class. As a by-product of these, we obtain various new families of chromatically equivalent graphs and chromatically unique graphs.

A new proof of a characterisation of Pfaffian bipartite graphs

In (J. Combin. Theory Ser. 18 (1975) 187) Little proved that a bipartite graph G is Pfaffian if and only if it does not contain an even subdivision H of K3.3 such that G - VH contains a l-factor. This paper gives a significantly shorter proof of this theorem.

Partition of a directed bipartite graph into two directed cycles

Abstract Let D = (V1, V2; A) be a directed bipartite graph with |V1| = |V2| = n ⩾ 2. Suppose that dD(x) + dD(y) ⩾ 3n + 1 for all x ϵ V1 and y ϵ V2. Then D contains two vertex-disjoint directed cycles of lengths 2n1 and 2n2, respectively, for any positive integer partition n = n1 + n2. Moreover, the condition is sharp for even n and nearly sharp for odd n.

A note on the chromaticity of some 2-connected ( n , n +3)-graphs

Abstract Let P ( G , λ ) denote the chromatic polynomial of a graph G. A graph G is chromatically unique if G ≅ H for any graph H such that P ( H , λ )= P ( G , λ ). This note corrects an error in the proof of the chromatic uniqueness of certain 2-connected graphs with n vertices and n +3 edges.

The Fixed-Point Problem

The goal of this chapter is to devise a method for approximating solutions of equations. This method is called fixed point iteration and is a process whereby a sequence of more and more accurate approximations is found. The convergence of this sequence to the desired solution is discussed. The procedure is then refined to give Newton’s method.

Limits of Functions

The concept of a limit of a function is introduced and theorems proved which assist in the calculation of such limits where they exist. The concept is then modified to give one-sided limits and infinite limits.

Taylor Polynomials and Taylor Series

Taylor polynomials are used to approximate values of functions at specified points. The error incurred is investigated by means of Taylor’s theorem. A method for ensuring that the approximation is accurate to within a specified error tolerance is illustrated. Taylor polynomials are then used to define Taylor series. Several techniques for finding these series are also discussed.