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- F M Dong, K M Koh, K L Teo, Ling Chia, Ru-Ying Liu, Yee-Hock Peng +6 others
- 2005

Preface For a century, one of the most famous problems in mathematics was to prove the four-colour theorem. This has spawned the development of many useful tools for solving graph colouring problems. In a paper in 1912, Birkhoff proposed a way of tackling the four-colour problem by introducing a function P (M, λ), defined for all positive integers λ, to be… (More)

- F M Dong, Canada K L Teo, C H C Little, M D Hendy
- 2006

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial Σ n k=1 α(G, k)(x) k of degree n, where (x) k = x(x − 1). .. (x − k +1) and α(G, k) is real for all k. The adjoint polynomial of G is defined to be Σ n k=1 α(G, k)µ k , where G is the complement of G. We find the zeros of the adjoint polynomials of paths and cycles.

- F M Dong, K L Teo, C H C Little, M D Hendy
- 2006

Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. We study the properties of two invariants under adjoint equivalence.

Let G = (V, E) be a 2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let P G (q) denote the chromatic polynomial of G. We show that (−1) n P G (q) > 0 for all 1 < q ≤ 1.2040.... This result is a corollary of a more general result that (−1) n Z G (q, w) > 0 for all 1 < q ≤ 1.2040...,… (More)

For a connected graph G and any non-empty S ⊆ V (G), S is called a weakly connected dominating set of G if the subgraph obtained from G by removing all edges each joining any two vertices in V (G) \ S is connected. The weakly connected domination number γ w (G) is defined to be the minimum integer k with |S| = k for some weakly connected dominating set S of… (More)

For any graph G, let W (G) be the set of vertices in G of degrees larger than 3. We show that for any bridgeless graph G, if W (G) is dominated by some component of G − W (G), then F (G, λ) has no roots in (1, 2), where F (G, λ) is the flow polynomial of G. This result generalizes the known result that F (G, λ) has no roots in (1, 2) whenever |W (G)| 2. We… (More)

Let G be a connected graph with vertex set V (G). A set S of vertices in G is called a weakly connected dominating set of G if (i) S is a dominating set of G and (ii) the graph obtained from G by removing all edges joining two vertices in V (G) \ S is connected. A weakly connected dominating set S of G is said to be minimum or a γ w-set if |S| is minimum… (More)

For any positive integers a, b, c, d, let R a,b,c,d be the graph obtained from the complete graphs K a , K b , K c and K d by adding edges joining every vertex in K a and K c to every vertex in K b and K d. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of R a,b,c,d is either a real number or a… (More)