We prove that there is a constantcsuch that, for each positive integerk, every (2k+ 1) × (2k+ 1) arrayAon the symbols (1,. . .,2k+1) with at mostc(2k+1) symbols in every cell, and each symbol repeated at mostc(2k+1) times in every row and column isavoidable; that is, there is a (2k+1) × (2k+1) Latin squareSon the symbols 1,. . .,2k+1 such that, for eachi,j∈ {1,. . .,2k+1}, the symbol in position (i,j) ofSdoes not appear in the corresponding cell inA. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integern, if each cell in ann×narrayBis assigned a set ofm≤ ρnsymbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability thatBis avoidable tends to 1 asn→ ∞.

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3, 4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide sufficient conditions for the existence of such a subgraph.

Abstract. A proper edge coloring of a graph with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is (a, b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)-biregular graphs have interval colorings and that all (3, 9)-biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)-biregular, (4, 6)-biregular and (4, 8)-biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings.

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.

We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most d − 1 edges of the d-dimensional hypercube Q d can be extended to a proper d-edge coloring of Q d . Additionally, we characterize which partial edge colorings of Q d with precisely d precolored edges are extendable to proper d-edge colorings of Q d , and consider some related edge precoloring extension problems of hypercubes.Note that the conjecture on distance-2 matchings in [6] is sharp both with respect to the distance between precolored edges, and in the sense that ∆(G) + 1 can in general not be replaced by ∆(G), even if any two precolored edges are at arbitrarily large distance from each other [6]. In [6], it is proved that this conjecture holds for e.g. bipartite multigraphs and subcubic multigraphs, and in [11] it is proved that a version of the conjecture with the distance increased to 9 holds for general graphs.However, for one specific family of graphs, the balanced complete bipartite graphs K n,n , the edge precoloring extension problem was studied far earlier than in the above-mentioned references.Here the extension problem corresponds to asking whether a partial Latin square can be completed to a Latin square. In this form the problem appeared already in 1960, when Evans [7] stated his now classic conjecture that for every positive integer n, if n−1 edges in K n,n have been (properly) colored, then the partial coloring can be extended to a proper n-edge-coloring of K n,n . This conjecture was solved for large n by Häggkvist [15] and later for all n by Smetaniuk [18], and independently by Andersen and Hilton [2]. Moreover, Andersen and Hilton [2] characterized which n × n partial Latin squares with exactly n non-empty cells are extendable.In this paper we consider the edge precoloring extension problem for the family of hypercubes. Although matching extendability and subgraph containment problems have been studied extensively for hypercubes, (see e.g. [19,9,20] and references therein) the edge precoloring extension problem for hypercubes seems to be a hitherto quite unexplored line of research. As in the setting of completing partial Latin squares (and unlike the papers [6, 11]) we consider only proper edge colorings of hypercubes Q d using exactly ∆(Q d ) colors.We prove that every proper edge precoloring of the d-dimensional hypercube Q d with at most d − 1 precolored edges is extendable to a d-edge coloring of Q d , thereby establishing an analogue of the positive resolution of Evans' conjecture. Moreover, similarly to [2] we also characterize which proper precolorings with exactly d precolored edges are not extendable to proper d-edge colorings of Q d . We also consider the cases when the precolored edges form an induced matching or one or two hypercubes of smaller dimension. The paper is concluded by a conjecture and some examples and...