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On Sheehan's Conjecture for Graphs with Symmetry
Author(s) -
Šajna Mateja,
Wagner Andrew
Publication year - 2015
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.21838
Subject(s) - mathematics , conjecture , combinatorics , automorphism , graph , graph automorphism , hamiltonian path , petersen graph , automorphism group , hamiltonian (control theory) , discrete mathematics , voltage graph , line graph , mathematical optimization
Sheehan's Conjecture states that every hamiltonian 4‐regular graph possesses a second Hamilton cycle. In this article, we verify Sheehan's Conjecture for 4‐regular graphs of order n whose automorphism group has size at least2 n 5 if either n ¬ ≡ 0 ( modα ) for all α ∈ { 3 , 4 , 5 } , or n = α m for α ∈ { 3 , 4 , 5 } and m is either an odd prime or a power of 2. We also give lower bounds on the size of the automorphism group and degree of a regular hamiltonian graph that guarantee existence of a second Hamilton cycle.