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 $n-7$.
References
- S. Arumugam, D. Froncek, N. Kamatchi, Distance magic graphs – a survey, J. Indones. Math. Soc. Special Edition (2011) 1–9.
- G. Chartrand, L. Lesniak, Graphs and Digraphs, Chapman and Hall, CRC, Fourth edition, 2005.
- D. Froncek, Fair incomplete tournaments with odd number of teams and large number of games, Congr. Numer. 187 (2007) 83–89.
- D. Froncek, Handicap distance antimagic graphs and incomplete tournaments, AKCE Int. J. Graphs Comb. 10(2) (2013) 119–127.
- D. Froncek, Handicap incomplete tournaments and ordered distance antimagic graphs, Congr. Numer. 217 (2013) 93–99.
- D. Froncek, P. Kovár, T. Kovárová, Fair incomplete tournaments, Bull. Inst. Combin. Appl. 48 (2006) 31–33.
- J. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 5 (# DS6) (2015) 43 pp.
- T. Harmuth, Ueber magische Quadrate und ähnliche Zahlenfiguren, Arch. Math. Phys. 66 (1881) 286–313.
- T. Harmuth, Ueber magische Rechtecke mit ungeraden Seitenzahlen, Arch. Math. Phys. 66 (1881) 413–447.
- M. Miller, C. Rodger, R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin. 28 (2003) 305–315.
- K. A. Sugeng, D. Froncek, M. Miller, J. Ryan, J. Walker, On distance magic labeling of graphs, J. Combin. Math. Combin. Comput. 71 (2009) 39–48.
- V. Vilfred, $sum$-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, Trivandrum, India, 1994.
Abstract
References
- S. Arumugam, D. Froncek, N. Kamatchi, Distance magic graphs – a survey, J. Indones. Math. Soc. Special Edition (2011) 1–9.
- G. Chartrand, L. Lesniak, Graphs and Digraphs, Chapman and Hall, CRC, Fourth edition, 2005.
- D. Froncek, Fair incomplete tournaments with odd number of teams and large number of games, Congr. Numer. 187 (2007) 83–89.
- D. Froncek, Handicap distance antimagic graphs and incomplete tournaments, AKCE Int. J. Graphs Comb. 10(2) (2013) 119–127.
- D. Froncek, Handicap incomplete tournaments and ordered distance antimagic graphs, Congr. Numer. 217 (2013) 93–99.
- D. Froncek, P. Kovár, T. Kovárová, Fair incomplete tournaments, Bull. Inst. Combin. Appl. 48 (2006) 31–33.
- J. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 5 (# DS6) (2015) 43 pp.
- T. Harmuth, Ueber magische Quadrate und ähnliche Zahlenfiguren, Arch. Math. Phys. 66 (1881) 286–313.
- T. Harmuth, Ueber magische Rechtecke mit ungeraden Seitenzahlen, Arch. Math. Phys. 66 (1881) 413–447.
- M. Miller, C. Rodger, R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin. 28 (2003) 305–315.
- K. A. Sugeng, D. Froncek, M. Miller, J. Ryan, J. Walker, On distance magic labeling of graphs, J. Combin. Math. Combin. Comput. 71 (2009) 39–48.
- V. Vilfred, $sum$-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, Trivandrum, India, 1994.
Details
| Journal Section | Articles |
|---|---|
| Authors | |
| Publication Date | August 9, 2016 |
| Published in Issue | Year 2016 Volume: 3 Issue: 3 |