Fengming Dong

On chromatic uniqueness of two infinite families of graphs

In this paper, it is proven that for each k ≥ 2, m ≥ 2, the graph Θk(m,…,m), which consists of k disjoint paths of length m with same ends is chromatically unique, and that for each m, n, 2 ≤ m ≤ n, the complete bipartite graph Km,n is chromatically unique.

Proof of a Chromatic Polynomial Conjecture

Let P(G, ?) denote the chromatic polynomial of a graph G. It is proved in this paper that for every connected graph G of order n and real number ??n, (??2)n?1P(G, ?)??(??1)n?2P(G, ??1)?0. By this result, the following conjecture proposed by Bartels and Welsh is proved: P(G, n)(P(G, n?1))?1>e for every graph G of order n.

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.

On the number of perfect matchings of line graphs

Let G=(V,E) be a connected graph, where |E| is even. In this paper we show that the line graph L(G) of G contains at least 2^|^E^|^-^|^V^|^+^1 perfect matchings, and characterize G such that L(G) has exactly 2^|^E^|^-^|^V^|^+^1 perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagome lattice, 3.12.12 lattice, and Sierpinski gasket with dimension two in the context of statistical physics.

A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

‡ Abstract. Let G ¼ð V;EÞ be a nonseparable plane graph on n vertices with at least two edges. Suppose that G has outer face C and that every 2-vertex-cut of G contains at least one vertex of C. Let PGðqÞ denote the chromatic polynomial of G. We show that ð−1Þ n PGðqÞ > 0 for all 1 0 for all 1 < q ≤ 1.2040:::, where ZGðq;wÞ is the multivariate Tutte polynomial of G, w ¼f wege∈E, we ¼ −1 for all e which are not incident to a vertex of C, we ∈ W2 for all e ∈ EðC Þ, we ∈ W1 for all other edges e, and W1, W2 are suitably chosen intervals with −1 ∈ W1 ⊂ W2 ⊆ ð−2;0Þ.

Determining the component number of links corresponding to lattices

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

The maximum number of maximal independent sets in unicyclic connected graphs

In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value.

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

For a graph

The largest non-integer real zero of chromatic polynomials of graphs with fixed order

G