22 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
An approximability-related parameter on graphs―-properties and applications
Graph TheoryInternational audienceWe introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col
Density of -critical signed graphs
We say that a signed graph is -critical if it is not -colorable but
every one of its proper subgraphs is -colorable. Using the definition of
colorability due to Naserasr, Wang, and Zhu that extends the notion of circular
colorability, we prove that every -critical signed graph on vertices has
at least edges, and that this bound is asymptotically tight.
It follows that every signed planar or projective-planar graph of girth at
least is (circular) -colorable, and for the projective-planar case, this
girth condition is best possible. To prove our main result, we reformulate it
in terms of the existence of a homomorphism to the signed graph ,
which is the positive triangle augmented with a negative loop on each vertex.Comment: 27 pages, 12 figure