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The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Catalogs of Regular Graphs
 Intern. J. Computer Math
, 1994
"... Exhaustive catalogs of graphs can be an indispensable tool for computerized investigations in graphtheoretical research. We describe an algorithm for the creation of catalogs of regular graphs. Using this algorithm, all of the regular graphs on thirteen or fewer vertices have been created. 1 Intro ..."
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Exhaustive catalogs of graphs can be an indispensable tool for computerized investigations in graphtheoretical research. We describe an algorithm for the creation of catalogs of regular graphs. Using this algorithm, all of the regular graphs on thirteen or fewer vertices have been created. 1 Introduction Exhaustive catalogs of graphs can be an invaluable tool for research in graph theory. To be able to verify a conjecture quickly for all small graphs gives one extra confidence, and sometimes extra insight, into a problem. Or the discovery of a counterexample can lead to a necessary change in the hypothesis of a theorem. Additionally, in attempting to prove a result, one can use experiments on catalogs to help suggest or refine elements of a proof. In this report we describe the generation of catalogs of regular graphs on thirteen or fewer vertices. The algorithm used is fairly simple, with the exception of the need for a routine to determine if two graphs are isomorphic. Basically,...