Restricted completion of sparse partial Latin squares

Abstract

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

“…The number of colors in Theorem 1 agrees with the chromatic index of the complete graph if p ∈ {4r, 4r − 1} and is thus best possible; we do not know whether t = 4r − 2 can be replaced by t = 4r − 3 if p ∈ {4r − 2, 4r − 3}. In fact, the number of colors used in Theorem 1 is due to the proof method used in this paper: the general proof method in the papers [9,5,4,7] rely on the existence of a proper edge coloring of the considered graph where every (or almost every) edge is contained in a large number of 2-colored 4-cycles. Roughly speaking, after applying a simple probabilistic argument, the idea is then to switch colors on such 4-cycles in a systematic way, so that the resulting coloring agrees with the precoloring and respects the colors forbidden by the list assignment.…”

“…Combining the notions of extending a sparse precoloring and avoiding a sparse list assignment, Andrén et al [5] proved that there are constants α > 0 and β > 0, such that for every positive integer n, every α-dense partial edge coloring of K n,n can be extended to a proper n-edge-coloring avoiding any given β-sparse list assignment L, provided that no edge e is precolored by a color that appears in L(e). Quite recently, Casselgren et al [9] obtained analogous results for hypercubes.…”

“…In particular, this means that a large number of edges in K 2n are not contained in 2-edgecolored 4-cycles, but every edge is adjacent to "many" edges that are contained in such 4-cycles. Nevertheless, to be able to apply the machinery from [5], we need to significantly strengthen these techniques.…”

Restricted extension of sparse partial edge colorings of hypercubes

“…Combining the notion of extending a precoloring and avoiding a list assignment, Andren et al [3] proved that every "sparse" partial edge coloring of K n,n can be extended to a proper n-edge coloring avoiding a given list assignment L satisfying certain "sparsity" conditions, provided that no edge e is precolored by a color that appears in L(e); we refer to [3] for the exact definition of "sparse" in this context. An analogous result for complete graphs was recently obtained in [8].…”

Latin Cubes with Forbidden Entries

“…If every cell contains at most m symbols, and every symbol occurs at most m times in every row and column, then A is an (m, m, m)-array. Confirming a conjecture by Häggkvist [11], it was proved in [1] that there is a constant c > 0 such that if m ≤ cn and A is an (m, m, m)array, then A is avoidable; that is, there is a Latin square L such that for every (i, j) the symbol in position (i, j) in L is not in A(i, j) (see also [3,2]). The purpose of this note is to prove an analogue of this result for Latin cubes of order n = 2 k .…”