Interval colorings of complete bipartite graphs and trees

In the work interval colorings [1] of complete bipartite graphs and trees are investigated. The obtained results were announced in [2]. Non defined concepts can be found in [3, 4]. Let G = (V (G), E(G)) be an undirected graph without multiple edges and loops. The degree of a vertex x in G is denoted by dG(x), the greatest degree of vertices – by ∆(G), the chromatic index of G – by χ(G). Interval t-coloring of a graph G is a proper coloring of edges of G by the colors 1, . . . , t, at which by each color i, 1 ≤ i ≤ t, at least one edge ei ∈ E(G) is colored, and edges incident with each vertex x ∈ V (G) are colored by dG(x) consecutive colors. A graph G is called interval colorable if there is t ≥ 1 for which G has an interval t-coloring. For an interval colorable graph G, we denote by w(G) and W (G), respectively, the least and the greatest value of t, for which G has an interval t-coloring. If α is a proper edge coloring of a graph G, then the color of an edge e ∈ E(G) at this coloring is denoted by α(e, G) or, if it is clear which graph is spoken about, by α(e). Let k and l be positive integers. Let us denote by σ(k, l) the greatest common divisor of k and l. The algorithm of Euclid for finding of σ(k, l) consists of the construction of sequences (Fi(k, l)), (fi(k, l)), i = 1, 2, . . ., defined as follows: F1(k, l) = max{k, l}, f1(k, l) = min{k, l}; if F1(k, l) = f1(k, l) then the construction of the sequences is finished, and if F1(k, l) > f1(k, l) then Fi+1(k, l) = max{Fi(k, l)−fi(k, l), fi(k, l)}, fi+1(k, l) = min{Fi(k, l)−fi(k, l), fi(k, l)}, i = 1, 2, . . .. The algorithm is completed at the finding of such j (let us denote it by s(k, l)) for which Fj(k, l) = fj(k, l) = σ(k, l). Let H(μ, ν) be a (0, 1)-matrix with μ rows, ν columns, and with elements hij, 1 ≤ i ≤ μ, 1 ≤ j ≤ ν. The i-th row of the matrix H(μ, ν), 1 ≤ i ≤ μ, is called collected, if hip = hiq = 1, p ≤ t ≤ q imply hit = 1, and the inequality ∑ν j=1 hij ≥ 1 holds. Similarly, the j-th column of the matrix H(μ, ν), 1 ≤ j ≤ ν, is called collected, if hpj = hqj = 1, p ≤ t ≤ q imply htj = 1, and the inequality ∑μ i=1 hij ≥ 1 holds. For the i-th row of the matrix H(μ, ν), all rows and columns of which are collected, define a number ε(i, H(μ, ν)) = minhij=1 j, i = 1, . . . , μ. For the j-th column of the matrix H(μ, ν), all rows and columns of which are collected, define a number ξ(j,H(μ, ν)) = |{i/ ε(i, H(μ, ν)) = j, 1 ≤ i ≤ μ}|, j = 1, . . . , ν. H(μ, ν) is called an r-regular (r ≥ 1) matrix, if ∑ν j=1 hij = r, i = 1, . . . , μ. H(μ, ν) is called a collected matrix, if all its rows and columns are collected, h11 = hμν = 1, and the inequality ε(1, H(μ, ν)) ≤ . . . ≤ ε(μ,H(μ, ν)) holds. (0, 1)-matrices A(α, γ) and B(β, γ) with elements aij , 1 ≤ i ≤ α, 1 ≤ j ≤ γ and bij , 1 ≤ i ≤ β, 1 ≤ j ≤ γ, respectively, are called equivalent, if ∑α i=1 aij = ∑β i=1 bij , j = 1, . . . , γ.