Eng Guan Tay

Optimal Orientations of Graphs and Digraphs: A Survey

Abstract. For a graph or digraph G, let be the family of strong orientations of G; and for any , we denote by d(D) the diameter of D. Define . In this paper, we survey the results obtained and state some problems and conjectures for the parameter .

Optimal orientations of products of paths and cycles

Abstract For a graph G, let D (G) be the family of strong orientations of G, d (G) = min {d(D)∣D ∈ D (G)} and p(G) = d (G) − d(G) , where d(G) and d(D) are the diameters of G and D respectively. In this paper we show that p(G) = 0 if G is a cartesian product of 1. (1) paths, and 2. (2) paths and cycles, which satisfy some mild conditions.

Making mathematics practical : an approach to problem solving

Mathematical Problem Solving Scheme of Work and Assessment of the Mathematics Practical Detailed Lesson Plans Scaffolding Suggestions, Solutions to the Problems and Assessment Notes.

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 optimal orientations of cartesian products of graphs (II): complete graphs and even cycles

Abstract For a graph G, let D (G) be the family of strong orientations of G. Define d ⇀ (G)= min {d(D) /D∈ D (G)} and ρ(G)= d ⇀ (G)−d(G), where d(D) (resp., d(G)) denotes the diameter of the digraph D (resp., graph G). Let G×H denote the cartesian product of the graphs G and H, K p the complete graph of order p and Cp the cycle of order p. In this paper, we show that ρ(K 2 ×C 2m )=2, ρ(K n ×C 2m )=1 for n=3,4,5,7, and ρ(Kn×C2m)=0 for most cases otherwise.

On optimal orientations of cartesian products with a bipartite graph

Abstract For a graph G, let D (G) be the family of strong orientations of G. Define d → (G)= min {d(D) | D∈ D (G)} and ρ(G)= d → (G)−d(G), where d(D) (resp., d(G)) denotes the diameter of the digraph D (resp., graph G). Let G×H denote the cartesian product of the graphs G and H. In this paper, we show that ρ(G×A1×A2×⋯×Ak)=0, where G is a bipartite graph fulfilling certain weak conditions and {A i | 1⩽i⩽k} is certain combination of graphs.

On optimal orientations of cartesian products of graphs (I)

Abstract For a graph G , let D (G) be the family of strong orientations of G , and define d ⇀ ( G ) = min ⁡ { d ( D ) | D ∈ D ( G ) } , where d ( D ) denotes the diameter of the digraph D . Let G × H denote the cartesian product of the graphs G and H . In this paper, we determine completely the values of d ⇀ ( K m × P n ) , d ⇀ ( K m × K n ) and d ⇀ ( K n × C 2 k + 1 ) , except d ⇀ ( K 3 × C 2 k + 1 ) , k ⩾ 2, where K n , P n and C n denote the complete graph, path and cycle of order n , respectively.

On optimal orientations of cartesian products of even cycles and paths

For a graph G, let D(G) be the family of strong orientations of G, and define d(G) = min{d(D)|D∈D(G)}, where d(D) is the diameter of the digraph D. In this paper, we evaluate the values of d(C 2n x P k ), where n ≥ 2 and k ≥ 2.

On Optimal Orientations of Cartesian Products of Trees

Abstract. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {Ti}, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏i=1nTi admits an (r, d)-invariant orientation provided that d(T1)≥d(T2)≥4 for n=2, and d(T1)≥5 and d(T2)≥4 for n≥3.

Relooking ‘Look Back’: a student's attempt at problem solving using Polya's model

Against the backdrop of half a century of research in mathematics problem solving, Pólyas last stage is especially conspicuous – by the scarcity of research on it! Much of the research focused on the first three stages (J.M. Francisco and C.A. Maher, Conditions for promoting reasoning in problem solving: Insights from a longitudinal study, J. Math. Behav. 24 (2005), pp. 361–372; J.A. Taylor and C. Mcdonald, Writing in groups as a tool for non-routine problem solving in first year university mathematics, Int. J. Math. Educ. Sci. Technol. 38(5) (2007), pp. 639–655.), with little or no successful attempts at following through with the subjects. In this article, we describe a case study of how the innovation of a ‘Practical Worksheet’ within a new paradigm of a ‘Mathematics Practical’ enabled a high-achieving student to push beyond getting a solution for a problem to extending, adapting and generalizing his solution. The findings from this study indicate promise in achieving the learning of Polyas model with notable success in the fourth stage, Look Back.