On Pell Numbers and \((k_1A_1, k_2A_2, k_3A_3)\)-Edge Colouring in Graphs

Abstract

In this paper, we introduce a special \((k_1A_1, k_2A_2, k_3A_3)\)-edge colouring of a graph. We shall show that for special graphs and special values of \(k_i\), \(i = 1, 2, 3\), the number of such colourings generalizes the well-known Pell numbers. Using this graph interpretation, we give a direct formula for the generalized Pell numbers. Moreover, we show some identities for these numbers.