Abstract:

A  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 the weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum\limits_{x_j\in N(x_i)}f(x_j)$) forms an increasing arithmetic progression with difference one. A graph $G$ is a {\em handicap distance antimagic graph} if it allows a handicap distance antimagic labeling. We construct $(n-7)$-regular handicap distance antimagic graphs for every order $n\equiv2\pmod4$ with a few small exceptions. This result complements results by Kov\'a\v{r}, Kov\'a\v{r}ov\'a, and Krajc~[P. Kov\'a\v{r}, T. Kov\'a\v{r}ov\'a, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than $n-7$.

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