Coloring Powers and Girth

International audienceAlon and Mohar (2002) posed the following problem: among all graphs G of maximum degree at most d and girth at least g, what is the largest possible value of χ(G t), the chromatic number of the tth power of G? For t ≥ 3, we provide several upper and lower bounds concerning this problem, all of which are sharp up to a constant factor as d → ∞. The upper bounds rely in part on the probabilistic method, while the lower bounds are various direct constructions whose building blocks are incidence structures. 1. Introduction. For a positive integer t, the tth power G t of a (simple) graph G = (V, E) is a graph with vertex set V in which two distinct elements of V are joined by an edge if there is a path in G of length at most t between them. What is the largest possible value of the chromatic number χ(G t) of G t , among all graphs G with maximum degree at most d and girth (the length of the shortest cycle contained in the graph) at least g? For t = 1, this question was essentially a long-standing problem of Vizing [11], one that stimulated much work on the chromatic number of bounded degree triangle-free graphs, and was eventually settled asymptotically by Johansson [6] using the probabilistic method. In particular, he showed that the largest possible value of the chromatic number over all girth 4 graphs of maximum degree at most d is Θ(d/ log d) as d → ∞. The case t = 2 was considered and settled asymptotically by Alon and Mohar [2]. They showed that the largest possible value of the chromatic number of a graph's square taken over all girth 7 graphs of maximum degree at most d is Θ(d 2 / log d) as d → ∞. Moreover, there exist girth 6 graphs of arbitrarily large maximum degree d such that the chromatic number of their square is (1 + o(1))d 2 as d → ∞. In this work, we consider this extremal question for larger powers t ≥ 3, which was posed as a problem in [2], and settle a range of cases for g. A first basic remark to make is that, ignoring the girth constraint, the maximum degree Δ(G t) of G t for G a graph of maximum degree at most d satisfie

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

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

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

On First-Order Definable Colorings

We address the problem of characterizing $H$-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give several characterizations of a homomorphism dualities arising in a class of structure