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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...
Small snarks with large oddness
"... We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph G with oddness ω(G) other than the Petersen graph has at least 5.41ω(G) vertices, and for each integer k with 2 6 k 6 6 we construct an infinite family of cu ..."
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We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph G with oddness ω(G) other than the Petersen graph has at least 5.41ω(G) vertices, and for each integer k with 2 6 k 6 6 we construct an infinite family of cubic graphs with cyclic connectivity k and small oddness ratio V (G)/ω(G). In particular, for cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of
The asymptotic distribution of the number of 3star factors in random dregular graphs
"... The Small Subgraph Conditioning Method has been used to study the almost sure existence and the asymptotic distribution of the number of regular spanning subgraphs of various types in random dregular graphs. In this paper we use the method to determine the asymptotic distribution of the number of ..."
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The Small Subgraph Conditioning Method has been used to study the almost sure existence and the asymptotic distribution of the number of regular spanning subgraphs of various types in random dregular graphs. In this paper we use the method to determine the asymptotic distribution of the number of 3star factors in random dregular graphs for d ≥ 4.