665 research outputs found

    The Incidence Chromatic Number of Toroidal Grids

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    An incidence in a graph GG is a pair (v,e)(v,e) with v∈V(G)v \in V(G) and e∈E(G)e \in E(G), such that vv and ee are incident. Two incidences (v,e)(v,e) and (w,f)(w,f) are adjacent if v=wv=w, or e=fe=f, or the edge vwvw equals ee or ff. The incidence chromatic number of GG is the smallest kk for which there exists a mapping from the set of incidences of GG to a set of kk colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n=Cm□CnT_{m,n}=C_m\Box C_n equals 5 when m,n≡0(mod5)m,n \equiv 0 \pmod 5 and 6 otherwise.Comment: 16 page

    Knapsack problem with objective value gaps

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    International audienceWe study a 0-1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems
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