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

Colouring exact distance graphs of chordal graphs

For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently, there has been an effort to obtain bounds on the chromatic number $\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which arise from reviewers' comment

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

Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

Injective colorings of sparse graphs

Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $\chi_i(G)\leq\Delta(G) + 1$; and if $mad(G) < 42/19$, then $\chi_i(G)=\Delta(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $\Delta(G)\geq 4$. We prove that if $g(G)\geq 9$, then $\chi_i(G)\leq\Delta(G)+1$; similarly, if $g(G)\geq 13$, then $\chi_i(G)=\Delta(G)$.Comment: 10 page