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Defective 2-colorings of sparse graphs. (English) Zbl 1282.05041
Summary: A graph \(G\) is \((j,k)\)-colorable if its vertices can be partitioned into subsets \(V_1\) and \(V_2\) such that every vertex in \(G[V_1]\) has degree at most \(j\) and every vertex in \(G[V_2]\) has degree at most \(k\). We prove that if \(k\geqslant 2j+2\), then every graph with maximum average degree at most \(2\left(2-\frac{k+2}{(j+2)(k+1)}\right)\) is \((j,k)\)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close to \(2\left(2-\frac{k+2}{(j+2)(k+1)}\right)\) (from above) that are not \((j,k)\)-colorable.In fact, we prove a stronger result by establishing the best possible sufficient condition for the \((j,k)\)-colorability of a graph \(G\) in terms of the minimum, \(\phi_{j,k}(G)\), of the difference \(\phi_{j,k}(W,G)=\left(2-\frac{k+2}{(j+2)(k+1)}\right)| W| -| E(G[W])|\) over all subsets \(W\) of \(V(G)\). Namely, every graph \(G\) with \(\phi_{j,k}(G)>\frac{-1}{k+1}\) is \((j,k)\)-colorable. On the other hand, we construct infinitely many non-\((j,k)\)-colorable graphs \(G\) with \(\phi_{j,k}(G)=\frac{-1}{k+1}\).

MSC:
05C15 Coloring of graphs and hypergraphs
05C42 Density (toughness, etc.)
05C35 Extremal problems in graph theory
05C07 Vertex degrees
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[1] Appel, K.; Haken, W., Every planar map is four colorable. part I. discharging, Illinois J. Math., 21, 429-490, (1977) · Zbl 0387.05009
[2] Appel, K.; Haken, W., Every planar map is four colorable. part II. reducibility, Illinois J. Math., 21, 491-567, (1977) · Zbl 0387.05010
[3] Borodin, O. V.; Ivanova, A. O., Near-proper List vertex 2-colorings of sparse graphs, Diskretn. Anal. Issled. Oper., J. Appl. Ind. Math., 4, 1, 21-23, (2010), (in Russian); English translation in:
[4] Borodin, O. V.; Kostochka, A. V., Vertex partitions of sparse graphs into an independent vertex set and subgraph of maximum degree at most one, Sibirsk. Mat. Zh., Sib. Math. J., 52, 5, 796-801, (2011), (in Russian); translation in: · Zbl 1232.05073
[5] Borodin, O. V.; Ivanova, A. O.; Montassier, M.; Ochem, P.; Raspaud, A., Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k, J. Graph Theory, 65, 83-93, (2010) · Zbl 1209.05177
[6] Borodin, O. V.; Ivanova, A. O.; Montassier, M.; Raspaud, A., \((k, j)\)-coloring of sparse graphs, Discrete Appl. Math., 159, 17, 1947-1953, (2011) · Zbl 1239.05059
[7] Borodin, O. V.; Ivanova, A. O.; Montassier, M.; Raspaud, A., \((k, 1)\)-coloring of sparse graphs, Discrete Math., 312, 6, 1128-1135, (2012) · Zbl 1238.05084
[8] Cowen, L. J.; Cowen, R. H.; Woodall, D. R., Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory, 10, 187-195, (1986) · Zbl 0596.05024
[9] Glebov, A. N.; Zambalaeva, D. Zh., Path partitions of planar graphs, Sib. Elektron. Mat. Izv., 4, 450-459, (2007), (in Russian) · Zbl 1132.05315
[10] Havet, F.; Sereni, J.-S., Improper choosability of graphs and maximum average degree, J. Graph Theory, 52, 181-199, (2006) · Zbl 1104.05026
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