Some recent discoveries about half-arc-transitive graphs

Abstract

We present some new discoveries about graphs that are half-arc-transitive (that is, vertex-and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest * The first author was supported by a James Cook Fellowship and a Marsden Fund grant (UOA1015) from the Royal Society of New Zealand with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser D 4 (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.

Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8

“…However, the full automorphism group of these graphs is in all but two cases larger than this guaranteed G and is in fact dart-transitive. The two exceptional graphs, where the 1 2 -arc-transitive group G equals the full automorphism group, have order 10752 and are precisely the two graphs constructed in [4] and [5]. This confirms Theorem 1.…”

“…Their example has order 10752 and vertex-stabiliser isomorphic to the dihedral group D 4 of order 8. In [5], two more examples with Aut(Γ) v ∼ = D 4 were found (another one of order 10752 and one of order 21870), and also the first known example of a tetravalent 1 2 -arc-transitive graph with a non-abelian vertex-stabiliser of order 16; the latter having order 90 · 3 10 . To the best of our knowledge, these four graphs are the only known tetravalent 1 2 -arc-transitive graphs with a non-abelian vertex-stabiliser.…”

Linking rings structures and semisymmetric graphs: Cayley constructions

“…It should be pointed out that the G-half-arc transitive case has received much attention in the past (see, for example [4,6,8,9,12,15]), so it is only natural to initiate a similar systematic study of the other…”

Tetravalent half‐arc‐transitive bi‐p‐metacirculants

“…Note that in his results, Bouwer provided just one graph for each even valency k 2 > 2, and this was recently improved by Conder and Žitnik in [7] and by Rivera and Šparl in [20] by showing that infinitely many HAT graphs of any given even valency greater than 2 exist. Following Bouwer's work, HAT graphs have received a great deal of attention, and constructing and characterizing tetravalent HAT graphs, especially in the 4-valent case, is also currently an active topic in algebraic graph theory (see, for instance, [1,[5][6][7]14,15,[17][18][19]21,22,27]).…”