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On 1-improper 2-coloring of sparse graphs. (English) Zbl 1281.05060
Summary: A graph \(G\) is \((1,1)\)-colorable if its vertices can be partitioned into subsets \(V_1\) and \(V_2\) such that every vertex in \(G[V_i]\) has degree at most 1 for each \(i\in\{1,2\}\). We prove that every graph with maximum average degree at most \(\frac{14}{5}\) is \((1,1)\)-colorable. In particular, it follows that every planar graph with girth at least 7 is (\(1,1\))-colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to \(\frac{14}{5}\) (from above) that are not \((1,1)\)-colorable.
In fact, we establish the best possible sufficient condition for the \((1,1)\)-colorability of a graph \(G\) in terms of the minimum, \(\rho_G\), of \(\rho_G(S)=7| S| -5| E(G[S])|\) over all subsets \(S\) of \(V(G)\). Namely, every graph \(G\) with \(\rho_G\geq 0\) is \((1,1)\)-colorable. On the other hand, we construct infinitely many non-\((1,1)\)-colorable graphs \(G\) with \(\rho_G=-1\). This solves a related conjecture of A. Kurek and A. Ruciński from [J. Graph Theory 18, No. 1, 73–81 (1994; Zbl 0790.05027)].

MSC:
05C15 Coloring of graphs and hypergraphs
05C42 Density (toughness, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C35 Extremal problems in graph theory
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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 vertex 2-colorings of sparse graphs, Diskretn. Anal. Issled. Oper., 16, 2, 16-20, (2009), (in Russian). Translated in: Journal of Applied and Industrial Mathematics 4 (1) (2010) 21-23 · Zbl 1249.05110
[4] Borodin, O. V.; Ivanova, A. O., List strong linear 2-arboricity of sparse graphs, J. Graph Theory, 67, 2, 83-90, (2011) · Zbl 1218.05038
[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., \((k, 1)\)-coloring of sparse graphs, Discrete Math., 312, 6, 1128-1135, (2012) · Zbl 1238.05084
[7] 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
[8] Borodin, O. V.; Kostochka, A. V., On an upper bound of a graph’s chromatic number, depending on the graph’s degree and density, J. Comb. Theory B, 23, 247-250, (1977) · Zbl 0336.05104
[9] 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., 52, 5, 1004-1010, (2011), (in Russian). Translation in: Siberian Mathematical Journal 52 (5) 796-801 · Zbl 1232.05073
[10] O.V. Borodin, A.V. Kostochka, Defective 2-colorings of sparse graphs, submitted for publication. · Zbl 1282.05041
[11] 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
[12] Erdős, P.; Rubin, A.; Taylor, H., Choosability in graphs, Congr. Numer., 26, 125-157, (1979)
[13] Gerencsér, L., Szinezesi problemacrol, Mat. Lapok, 16, 274-277, (1965)
[14] Glebov, A. N.; Zambalaeva, D. Zh., Path partitions of planar graphs, Sib. Elektron. Mat. Izv., 4, 450-459, (2007), (in Russian) · Zbl 1132.05315
[15] Havet, F.; Sereni, J.-S., Improper choosability of graphs and maximum average degree, J. Graph Theory, 52, 181-199, (2006) · Zbl 1104.05026
[16] Kurek, A.; Ruciński, A., Globally sparse vertex-Ramsey graphs, J. Graph Theory, 18, 1, 73-81, (1994) · Zbl 0790.05027
[17] Lovász, L., On decomposition of graphs, Studia Sci. Math. Hungar, 1, 237-238, (1966) · Zbl 0151.33401
[18] Mihók, P., On vertex partition numbers of graphs, (Graphs and Other Combinatorial Topics (Prague, 1982), Teubner-Texte Math., vol. 59, (1983), Teubner Leipzig), 183-188
[19] Vizing, V. G., Vertex colourings with given colours, Metody Discret. Analiz, 29, 3-10, (1976), (in Russian)
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