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The maximum valency of regular graphs with given order and odd girth
Author(s) -
Zhang GuoHui
Publication year - 1992
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190160303
Subject(s) - mathematics , combinatorics , odd graph , valency , girth (graph theory) , graph , conjecture , discrete mathematics , order (exchange) , 1 planar graph , chordal graph , linguistics , philosophy , finance , economics
The odd girth of a graph G is the length of a shortest odd cycle in G . Let d ( n, g ) denote the largest k such that there exists a k ‐regular graph of order n and odd girth g . It is shown that d n, g ≥ 2| n / g ≥ if n ≥ 2 g . As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2 j ‐regular graph of order n and odd girth g if and only if n ≥ gj , where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.