F. M. Dong

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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)
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)
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)