On m-Bonacci-Sum Graphs

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.