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.

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.

We consider a semi-online version of the problem of scheduling a sequence of jobs of different lengths on two uniform machines with given speeds 1 and s. Jobs are revealed one by one (the assignment of a job has to be done before the next job is revealed), and the objective is to minimize the makespan. In the considered variant the optimal offline makespan is known in advance. The most studied question for this online-type problem is to determine the optimal competitive ratio, that is, the worst-case ratio of the solution given by an algorithm in comparison to the optimal offline solution. In this paper, we make a further step towards completing the answer to this question by determining the optimal competitive ratio for s between 5+ √ 241 12 ≈ 1.7103 and √ 3 ≈ 1.7321, one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds.

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