79 research outputs found
On Colorings of Graph Powers
In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a -cycle
exists if and only if the chromatic number of the st power of
is less than or equal to 3, where is the 2-subdivision of . We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented
Complements of nearly perfect graphs
A class of graphs closed under taking induced subgraphs is -bounded if
there exists a function such that for all graphs in the class, . We consider the following question initially studied in [A.
Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em
Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a
-bounded class , is the class -bounded (where
is the class of graphs formed by the complements of graphs from
)? We show that if is -bounded by the constant function
, then is -bounded by
and this is best possible. We show that for
every constant , if is -bounded by a function such that
for , then is -bounded. For every ,
we construct a class of graphs -bounded by whose
complement is not -bounded
A Fixed-Parameter Algorithm for the Schrijver Problem
The Schrijver graph is defined for integers and with as the graph whose vertices are all the -subsets of
that do not include two consecutive elements modulo , where two such sets
are adjacent if they are disjoint. A result of Schrijver asserts that the
chromatic number of is (Nieuw Arch. Wiskd., 1978). In the
computational Schrijver problem, we are given an access to a coloring of the
vertices of with colors, and the goal is to find a
monochromatic edge. The Schrijver problem is known to be complete in the
complexity class . We prove that it can be solved by a randomized
algorithm with running time , hence it is
fixed-parameter tractable with respect to the parameter .Comment: 19 pages. arXiv admin note: substantial text overlap with
arXiv:2204.0676
Choice number of Kneser graphs
In this short note, we show that for any and
the choice number of the Kneser graph is
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