Regular handicap tournaments of high degree
Dalibor Fronček, Aaron Shepanik · Journal of Algebra Combinatorics Discrete Structures and Applications · 2016
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 $n-7$.