Local coloring of Kneser graphs

Abstract

A local coloring of a graph G is a function c : V (G) → N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ m S , where m S is the number of edges of the induced subgraph S . The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K (n, k) for some values of n and k is determined.

Local colourings of Cartesian product graphs

“…In Section 3, the local chromatic number is determined for Cartesian products of 3-colourable graphs. In particular, the local chromatic number of products of cycles is extracted, a result that in one part corrects an assertion from [7]. In the final section we then prove that if G and H are graphs such that χ(G) ≤ χ (H)/2 , then χ (G H) ≤ χ (H) + 1.…”

“…In the subsequent paper [1], the emphasize was on regular graphs, where Cartesian products with one factor being a hypercube played the central role. In [7] it was proved that χ (G) ≤ 2 (G) − 2 holds for any graph with (G) ≥ 3 and different from K 4 and K 5 . This result in particular confirms Conjecture 4.2 from [2] asserting that χ (G) = 4 holds for cubic, non-bipartite, and non-complete graphs.…”

“…This result in particular confirms Conjecture 4.2 from [2] asserting that χ (G) = 4 holds for cubic, non-bipartite, and non-complete graphs. In [8] local colourings of Kneser graphs were studied, this line or research was continued in [4] through the perceptive of semi-matching colourings.…”

“…In this note we are interested in the local colouring of Cartesian products of graphs, primarily motivated with investigations in [7] where exact local chromatic numbers were determined for some specific products. We proceed as follows.…”

Graph coloring with cardinality constraints on the neighborhoods

“…It is also close to the so-called star coloring problem studied in [9], and to the frugal coloring problem studied in [10]. Related work has been carried out recently by several authors (see [11][12][13][14][15][16]) including dramatic applications of coloring (see [17]). …”