Abstract
An equalized incomplete tournament EIT(p, r) on p teams which are ranked from 1 to p, is a tournament in which every team plays against r teams and the total strength of the opponents that every team plays with is a constant. A handicap incomplete tournament HIT(p, r) on p teams is a tournament in which every team plays against r opponents in such a way that
-
(i)
the total strength of the opponents that the stronger teams play with are higher, and
-
(ii)
the total strength of the opponents that the weaker teams play with are lower.
Thus, every team has an equal chance of winning in a HIT(p, r). A d-handicap labeling of a graph \(G=(V,E)\) on p vertices is a bijection \(l:V\rightarrow \{1,2,\cdot \cdot \cdot ,p\}\) with the property that \(l(v_i)=i\) and the sequence of weights \(w(v_1),w(v_2),\cdot \cdot \cdot ,w(v_p)\) forms an increasing arithmetic progression with difference d, where \(w(v_i) = \sum _{v\in N(v_i)}l(v)\). A graph G is d-handicap graph if it admits a d-handicap labeling. Thus, existence of an r-regular d-handicap graph guarantees the existence of a HIT(p, r). In this paper, we give a method to construct new \((d+k)\)-handicap graphs from d-handicap graphs for all \(k\ge 1\) and as an application, we characterize the d-handicap labeling of Hamming graphs. Further, we give another method to construct EIT(p, r) from an infinite class of HIT(p, r) by increasing the number of rounds in HIT(p, r).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. CRC Press, Boca Raton (2011)
Sedláček, J.: Problem 27. Theory of graphs and its applications. (Smolenice 1963), pp. 163–164. Publication House of the Czechoslovak Academy of Sciences, Prague (1964)
Colbourn, C., Dinitz, J.: The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton (2007)
Bloom, G.S., Golomb, S.W.: Applications of numbered undirected graphs. Proc. IEEE 65, 562–570 (1977)
Bloom, G.S., Golomb, S.W.: Numbered complete graphs, unusual rulers, and assorted applications. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 53–65. Springer, Berlin (1978). https://doi.org/10.1007/BFb0070364
Froncek, D., Kovár, P., Kovárová, T.: Fair incomplete tournaments. Bull. Inst. Comb. Appl. 48, 31–33 (2006)
Froncek, D.: Fair incomplete tournaments with odd number of teams and large number of games. In: Proceedings of the Thirty-Eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 187, 83–89 (2007)
Vilfred, V.: \(\sum -\)labelled graphs and circulant graphs. Ph.D. thesis, University of Kerala, Trivandrum, India (1994)
Miller, M., Rodger, C., Simanjuntak, R.: Distance magic labelings of graphs. Australas. J. Comb. 28, 305–315 (2003)
Arumugam, S., Froncek, D., Kamatchi, N.: Distance magic graphs-a survey. J. Indones. Math. Soc. Special Edition 11–26 (2011)
Froncek, D.: Handicap distance antimagic graphs and incomplete tournaments. AKCE J. Graphs Comb. 10, 119–127 (2013)
Freyberg, B.: Distance magic-type and distance antimagic-type labelings of graphs. Ph.D. thesis, Michigan Technological University, Houghton, MI, USA (2017)
Kovár, P., Kovárová, T., Krajc, B., Kravčenko, M., Shepanik, A., Silber, A.: On regular handicap graphs of even order. In: 9th International Workshop on Graph Labelings (IWOGL 2016). Electron. Notes Discrete Math. 60, 69–76 (2017)
Froncek, D.: Regular handicap graphs of odd order. J. Comb. Math. Comb. Comput. 102, 253–266 (2017)
Froncek, D.: Incomplete tournaments and handicap distance antimagic graphs. Congr. Numer. 217, 93–99 (2013)
Froncek, D.: Full spectrum of regular incomplete 2-handicap tournaments of order \(n\equiv 0(mod16)\). J. Comb. Math. Comb. Comput. 106, 175–184 (2018)
Froncek, D.: A note on incomplete regular tournaments with handicap two of order \(n\equiv 8(mod16)\). Opusc. Math. 37, 557–566 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Prajeesh, A.V., Paramasivam, K., Kamatchi, N. (2019). A Note on Handicap Incomplete Tournaments. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-25005-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25004-1
Online ISBN: 978-3-030-25005-8
eBook Packages: Computer ScienceComputer Science (R0)