819 research outputs found
On-line list colouring of random graphs
In this paper, the on-line list colouring of binomial random graphs G(n,p) is
studied. We show that the on-line choice number of G(n,p) is asymptotically
almost surely asymptotic to the chromatic number of G(n,p), provided that the
average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log
n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that
if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is
larger than the chromatic number by at most a multiplicative factor of C, where
C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice
number is by at most a multiplicative constant factor larger than the chromatic
number
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
Independent transversals in locally sparse graphs
Let G be a graph with maximum degree \Delta whose vertex set is partitioned
into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G)
containing exactly one vertex from each part V_i. If it is also an independent
set, then we call it an independent transversal. The local degree of G is the
maximum number of neighbors of a vertex v in a part V_i, taken over all choices
of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all
part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta),
then G has an independent transversal for sufficiently large \Delta. This
extends several previous results and settles (in a stronger form) a conjecture
of Aharoni and Holzman. We then generalize this result to transversals that
induce no cliques of size s. (Note that independent transversals correspond to
s=2.) In that context, we prove that parts of size |V_i| >=
(1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence
of such a transversal, and we provide a construction that shows this is
asymptotically tight.Comment: 16 page
Bipartite induced density in triangle-free graphs
We prove that any triangle-free graph on vertices with minimum degree at
least contains a bipartite induced subgraph of minimum degree at least
. This is sharp up to a logarithmic factor in . Relatedly, we show
that the fractional chromatic number of any such triangle-free graph is at most
the minimum of and as . This is
sharp up to constant factors. Similarly, we show that the list chromatic number
of any such triangle-free graph is at most as
.
Relatedly, we also make two conjectures. First, any triangle-free graph on
vertices has fractional chromatic number at most
as . Second, any triangle-free
graph on vertices has list chromatic number at most as
.Comment: 20 pages; in v2 added note of concurrent work and one reference; in
v3 added more notes of ensuing work and a result towards one of the
conjectures (for list colouring
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