We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colors needed to color the plane with no two points at distance 1 having the same color. Exoo considered a more general problem concerning graphs G [a,b] with R 2 as the vertex set and two vertices adjacent if their distance is in the interval [a, b]. Exoo conjectured χ(G [a,b] ) = 7 for sufficiently small but positive difference between a and b. We partially answer this conjecture by proving that χ(G [a,b] )is an assignment of j-elemental sets of colors to the vertices of G, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number χ f (G) is the infimum of fractions k/j for j-fold coloring of G using k colors. We generalize a method by Hochberg and O'Donnel (who proved that G [1,1] 4.36) for the fractional coloring of graphs G [a,b] , obtaining a bound dependent on a b . We also present few specific and two general methods for j-fold coloring of G [a,b] for small j, in particular for G [1,1] and G [1,2] . The j-fold coloring for small j has strong practical motivation especially in scheduling theory, while graph G [1,2] is often used to model hidden conflicts in radio networks.

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In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper r-coloring ϕ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is "the most similar" to ϕ, i.e., the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any fixed r ≥ 3, even for bipartite planar graphs. Moreover, it is W [1]-hard even for bipartite graphs, when parameterized by the number k of allowed recoloring transformations. On the other hand, the problem is fixed-parameter tractable, when parameterized by k and the number r of colors.We provide a 2 n ⋅n O(1) algorithm for the problem and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the fixing number of a graph G. It is the minimum k such that k recoloring transformations are sufficient to transform any coloring of G into a proper one.

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In this paper we give a family of on-line algorithms for the classical coloring problem and the L(2, 1)-labeling of unit disc intersection graphs. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from an application of a fractional and a b-fold coloring of the plane. Moreover, we give an off-line algorithm improving the bound of the L(2, 1)-span of unit disk intersection graphs in terms of the maximum degree.