Abstract
It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745–756] that for \(n\ge 5\) every simple graph of order n and size at least \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n+5\) has an orientation of diameter two. We prove this conjecture and hence determine for every \(n\ge 5\) the minimum value of m such that every graph of order n and size m has an orientation of diameter two.
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References
Bau, S., Dankelmann, P.: Diameter of orientations of graphs with given minimum degree. Eur. J. Comb. 49, 126–133 (2015)
Chvátal, V., Thomassen, C.: Distances in orientations of graphs. J. Comb. Theory Ser. B 24(1), 61–75 (1978)
Dankelmann, P., Guo, Y., Surmacs, M.: Oriented diameter of graphs with given maximum degree. J. Graph Theory 88(1), 5–17 (2018)
Egawa, Y., Iida, T.: Orientation of \(2\)-edge-connected graphs with diameter \(3\). Far East J. Appl. Math. 26(2), 257–280 (2007)
Füredi, Z., Horák, P., Pareek, C.M., Zhu, X.: Minimal oriented graphs of diameter \(2\). Graphs Comb. 14, 345–350 (1998)
Koh, K.M., Tan, B.P.: The diameter of an orientation of a complete multipartite graph. Discrete Math. 149, 131–139 (1996)
Koh, K.M., Tan, B.P.: The minimum diameter of orientations of complete multipartite graphs. Graphs Comb. 12, 333–339 (1996)
Koh, K.M., Tay, E.G.: Optimal orientations of graphs and digraphs, a survey. Graphs Comb. 18(4), 745–756 (2002)
Kwok, P.K., Liu, Q., West, D.B.: Oriented diameter of graphs with diameter \(3\). J. Comb. Theory Ser. B 100(3), 265–274 (2010)
Robbins, H.E.: A theorem on graphs, with an application to a problem on traffic control. Am. Math. Mon. 46, 281–283 (1939)
Surmacs, M.: Improved bound on the oriented diameter of graphs with given minimum degree. Eur. J. Comb. 59, 187–191 (2017)
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The first author was supported by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina. The third author was supported in part by the National Research Foundation of South Africa, grant number 103553. The fourth author was supported in part by the NSF DMS, grant number 1600811.
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Cochran, G., Czabarka, É., Dankelmann, P. et al. A Size Condition for Diameter Two Orientable Graphs. Graphs and Combinatorics 37, 527–544 (2021). https://doi.org/10.1007/s00373-020-02261-x
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DOI: https://doi.org/10.1007/s00373-020-02261-x