A note on incomplete regular tournaments with handicap two of order n≡8(mod 16)

Abstract

A $d$-handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection $f:V\to {1,2,\ldots ,n}$ with the property that $f(x_i)=i$ and the sequence of weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum_{x_i x_j\in E}f(x_j)$) forms an increasing arithmetic progression with common difference $d$. A graph $G$ is a $d$-handicap distance antimagic graph if it allows a $d$-handicap distance antimagic labeling. We construct a class of $k$-regular $2$-handicap distance antimagic graphs for every order $n\equiv8\pmod{16}$, $n\geq 56$ and $6\leq k\leq n-50$.