488 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
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
Light subgraphs in graphs with average degree at most four
A graph is said to be {\em light} in a family of graphs if
at least one member of contains a copy of and there exists
an integer such that each member of
with a copy of also has a copy of such that
for all . In this
paper, we study the light graphs in the class of graphs with small average
degree, including the plane graphs with some restrictions on girth.Comment: 12 pages, 18 figure
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