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 $(2t+1)$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 $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. 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

Graph colourings using structured colour sets

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Colorings of complements of line graphs

Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph $\operatorname{KG}(n,2)$ that have a chromatic number equal to its chromatic number, namely $n-2$. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the $\operatorname{NP}$-hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this paper are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results