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\(r\)-hued coloring of sparse graphs. (English) Zbl 1380.05056

Summary: For two positive integers \(k, r\), a \((k, r)\)-coloring (or \(r\)-hued \(k\)-coloring) of a graph \(G\) is a proper \(k\)-vertex-coloring such that every vertex \(v\) of degree \(d_G(v)\) is adjacent to at least \(\min \{d_G(v), r \}\) distinct colors. The \(r\)-hued chromatic number, \(\chi_r(G)\), is the smallest integer \(k\) for which \(G\) has a \((k, r)\)-coloring. The maximum average degree of \(G\), denoted by \(\operatorname{mad}(G)\), equals \(\max \{2 | E(H) | / | V(H) | : H \text{ is a subgraph of } G \}\).
In this paper, we prove the following results using the well-known discharging method. For a graph \(G\), if \(\operatorname{mad}(G) < \frac{12}{5}\), then \(\chi_3(G) \leq 6\); if \(\operatorname{mad}(G) < \frac{7}{3}\), then \(\chi_3(G) \leq 5\); if \(G\) has no \(C_5\)-components and \(\operatorname{mad}(G) < \frac{8}{3}\), then \(\chi_2(G) \leq 4\).

MSC:

05C15 Coloring of graphs and hypergraphs
05C42 Density (toughness, etc.)
05C35 Extremal problems in graph theory
05C07 Vertex degrees
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