The main result of the papzer is that any planar graph with odd girth at least 10k À 7 has a homomorphism to the Kneser graph G 2k‡1 k , i.e. each vertex can be colored with k colors from the set f1; 2;. .. ; 2k ‡ 1g so that adjacent vertices have no colors in common. Thus, for example, if the odd girth of a planar graph is at least 13, then the graph has a homomorphism to G 5 2 , also known as...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

Let G = (V,E) be a graph on n vertices and f : V → [1, n] a one to one map of V onto the integers 1 through n. Let dilation(f) = max{|f(v)− f(w)| : vw ∈ E}. Define the bandwidth B(G) of G to be the minimum possible value of dilation(f) over all such one to one maps f . Next define the Kneser Graph K(n, r) to be the graph with vertex set ( [n] r ) , the collection of r-subsets of an n element se...

The Kneser graph K(n, k) is the graph whose vertices are the k-elements subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(2k + 1, k) is an...

The Kneser graph K (n, k) is the graph whose vertices are the k-element subsets of an n elements set, with two vertices adjacent if they are disjoint. The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K (n, k) is an intere...

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G respectively. Let G×H be the direct product graph of graphs G and H . We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G ×H) = max{α(G)|V (H)|, α(H)|V (G)|} and χ(G×H) = min{χ(G), χ(H)}. AMS Classification: 05C15, 05C69.

A proper edge coloring of a graph is strong if it creates no bichromatic path length three. It well known that for k $k$ -regular at least 2 − 1 $2k-1$ colors are needed. We show admits with and only covers the Kneser K ( , ) $K(2k-1,k-1)$ . In particular, cubic strongly 5-edge-colorable whenever Petersen graph. One implications this result conjecture about colorings subcubic graphs proposed by...

We give a simple combinatorial description of an (n−2k+2)-chromatic edge-critical subgraph the Schrijver graph SG(n,k), itself induced vertex-critical Kneser KG(n,k). This extends main result Kaiser and Stehlík (2020) [5] to all values k, sharpens classical results Lovász from 1970s.