Morris, Joy

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Browsing Morris, Joy by Subject "CI graphs"

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    Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs
    (Centre for Discrete Mathematics and Computing, 2024) Morris, Dave W.; Morris, Joy
    A finite group G is said to be a non-DCI group if there exist subsets S1 and S2 of G, such that the associated Cayley digraphs C−→ay(G; S1) and C−→ay(G; S2) are isomorphic, but no automorphism of G carries S1 to S2. Furthermore, G is said to be a non-CI group if the subsets S1 and S2 can be chosen to be closed under inverses, so we have undirected Cayley graphs Cay(G; S1) and Cay(G; S2). We show that if p is a prime number, and the elementary abelian p- group (Zp)r is a non-DCI group, then (Zp)r+3 is a non-CI group. In most cases, we can also show that (Zp)r+2 is a non-CI group. In particular, from Pablo Spiga’s proof that (Z3)8 is a non-DCI group, we conclude that (Z3)10 is a non-CI group. This is the first example of a non-CI elementary abelian 3-group.
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    ℤ_3^8 is not a CI-group
    (University of Primorska, 2024) Morris, Joy
    A Cayley graph Cay(G, S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G, T), there is a group automorphism α of G such that Sα = T. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group G is a CI-group if every Cayley graph on G has the CI property, and is a DCI-group if every Cayley digraph on G has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group. In 2009, Spiga showed that ℤ38 is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that ℤ35 is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian 3-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that ℤ310 is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order 38 described by Spiga is does not have the CI property, so ℤ38 is not a CI-group.