Abstract
AbstractThe degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the degree/geodecity problem concerns the smallest order of a k-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any k-geodetic mixed graph with excess one must have geodetic girth two and be totally regular, thereby proving an earlier conjecture of the authors.
Funder
London Mathematical Society
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Theoretical Computer Science
Reference40 articles.
1. Araujo-Pardo, G., Balbuena, C., Miller, M., Ždímalová, M.: A family of dense mixed graph of diameter 2. Discret. Appl. Math. 218, 57–63 (2017)
2. Araujo-Pardo, G., Hernàndez-Cruz, C., Montellano-Ballesteros, J.J.: Mixed cages. Graphs Comb. 35(5), 989–999 (2019)
3. Bannai, E., Ito, T.: Regular graphs with excess one. Discret. Math. 37(2–3), 147–158 (1981)
4. Bondy, J.A., Murty, U.S.R.: Graph theory with applications. (Vol. 290) London: Macmillan (1976)
5. Bosák, J.: Geodetic graphs. Combinatorics, in Proceedings of Colloquim, Keszthely, North-Holland, Amsterdam, 151–172 (1978)
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