Approximability Distance in the Space of H-Colourability Problems
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Nonblocker in h-minor free graphs: kernelization meets discharging
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
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Let $G$ be a graph and let $c: V(G)\to\binom{1,\ldots,5}{2}$ be an assignment of $2$-element subsets of the set $1,\ldots,5$ to the vertices of $G$ such that for every edge $vw$, the sets $c(v)$ and $c(w)$ are disjoint. We call such an assignment a $(5,2)$-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph $G$ is the length of the shortest odd cycle in $G$ ($\infty$ if $G$ is bipartite). We prove that every planar graph of odd-girth at least $9$ is $(5,2)$-colorable, and thus it is homomorphic to the Petersen graph. Also, this implies that such graphs have a fractional chromatic number at most $5\over2$. As a special case, this result holds for planar graphs of girth at least $8$.