201 research outputs found
Complexity of C_k-Coloring in Hereditary Classes of Graphs
For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs.
We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw
Fine structure of 4-critical triangle-free graphs III. General surfaces
Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of
triangle-free graphs on surfaces with respect to 3-coloring. Their description
however contains two substructures (both related to graphs embedded in plane
with two precolored cycles) whose coloring properties are not entirely
determined. In this paper, we fill these gaps.Comment: 15 pages, 1 figure; corrections from the review process. arXiv admin
note: text overlap with arXiv:1509.0101
Precoloring co-Meyniel graphs
The pre-coloring extension problem consists, given a graph and a subset
of nodes to which some colors are already assigned, in finding a coloring of
with the minimum number of colors which respects the pre-coloring
assignment. This can be reduced to the usual coloring problem on a certain
contracted graph. We prove that pre-coloring extension is polynomial for
complements of Meyniel graphs. We answer a question of Hujter and Tuza by
showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which
also generalizes results of Hujter and Tuza and of Hertz. Moreover we show
that, given a co-Meyniel graph, the corresponding contracted graph belongs to a
restricted class of perfect graphs (``co-Artemis'' graphs, which are
``co-perfectly contractile'' graphs), whose perfectness is easier to establish
than the strong perfect graph theorem. However, the polynomiality of our
algorithm still depends on the ellipsoid method for coloring perfect graphs
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