On m-Bonacci-Sum Graphs

TL;DR: The notion of m-bonacci-sum graphs denoted by \(G_{m,n}\) for positive integers m, n are introduced and it is shown that this graph is bipartite and for \(n\ge 2^{m-2}\), \(\{1,2,\ldots ,n\) has exactly \((m-1)\) components.

Abstract: We introduce the notion of m-bonacci-sum graphs denoted by \(G_{m,n}\) for positive integers m, n. The vertices of \(G_{m,n}\) are \(1,2,\ldots ,n\) and any two vertices are adjacent if and only if their sum is an m-bonacci number. We show that \(G_{m,n}\) is bipartite and for \(n\ge 2^{m-2}\), \(G_{m,n}\) has exactly \((m-1)\) components. We also find the values of n such that \(G_{m,n}\) contains cycles as subgraphs. We also use this graph to partition the set \(\{1,2,\ldots ,n\}\) into \(m-1\) subsets such that each subset is ordered in such a way that sum of any 2 consecutive terms is an m-bonacci number.