The minimum number of vertices with girth 6 and degree ... - CiteSeerX
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The minimum number of vertices with girth 6 and degree ... - CiteSeerX
aDepartment of Computer Science, Dalian University of Technology, Dalian 116024, ... vertices in the (D; g)-cage by f(D; g), which has the following properties.
Discrete Mathematics 269 (2003) 249 – 258
www.elsevier.com/locate/disc
The minimum number of vertices with girth 6 and degree set D = {r; m} Yang Yuanshenga;∗ , Weifa Liangb a Department
of Computer Science, Dalian University of Technology, Dalian 116024, People’s Republic of China b Department of Computer Science, Australian National University, Canberra ACT 0200, Australia Received 29 June 2001; received in revised form 28 March 2002; accepted 22 July 2002
Abstract A (D; g)-cage is a graph having the minimum number of vertices, with degree set D and girth g. Denote by f(D; g) the number of vertices in a (D; g)-cage. In this paper it is shown that f({r; m}; 6) ¿ 2(rm − m + 1) for any 2 6 r ¡ m, and f({r; m}; 6) = 2(rm − m + 1) if either (i) 2 6 r 6 5 and r ¡ m or (ii) m − 1 is a prime power and 2 6 r ¡ m. Upon these results, it is conjectured that f({r; m}; 6) = 2(rm − m + 1) for any r with 2 6 r ¡ m. c 2002 Elsevier B.V. All rights reserved. Keywords: Cage; Girth; Degree set; Symmetric graph
1. Introduction A (v; g)-cage is a graph having the minimum number of vertices, with valence v and girth g. The existence of (v; g)-cages was proved by Erd
