12,625 research outputs found
Locally -distance transitive graphs
We give a unified approach to analysing, for each positive integer , a
class of finite connected graphs that contains all the distance transitive
graphs as well as the locally -arc transitive graphs of diameter at least
. A graph is in the class if it is connected and if, for each vertex ,
the subgroup of automorphisms fixing acts transitively on the set of
vertices at distance from , for each from 1 to . We prove that
this class is closed under forming normal quotients. Several graphs in the
class are designated as degenerate, and a nondegenerate graph in the class is
called basic if all its nontrivial normal quotients are degenerate. We prove
that, for , a nondegenerate, nonbasic graph in the class is either a
complete multipartite graph, or a normal cover of a basic graph. We prove
further that, apart from the complete bipartite graphs, each basic graph admits
a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a
biquasiprimitive action. These results invite detailed additional analysis of
the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
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