2008
DOI: 10.1016/j.disc.2007.11.002
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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 intr… Show more

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Cited by 13 publications
(18 citation statements)
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“…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.…”
Section: Introductionmentioning
confidence: 63%
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“…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.…”
Section: Introductionmentioning
confidence: 63%
“…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.…”
Section: Introductionmentioning
confidence: 97%
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“…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]). …”
Section: Introductionmentioning
confidence: 92%