Avoiding Arrays of Odd Order by Latin Squares

Abstract

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→ ∞.

Coloring Complete and Complete Bipartite Graphs from Random Lists

“…, n})-list assignment 1 for L(K n,n ), then whp there is an L-coloring of L(K n,n ). Note that in [1] this result is formulated in the language of arrays and Latin squares.…”

On the threshold problem for Latin boxes

“…This theorem is proved in Section 2. It is actually an easy consequence of a stronger result of Andrén, Casselgren, and Öhman [2], who showed that an analogous minimum-degree result holds. We include it here because the proof is short and elegant.…”

Coloring graphs of various maximum degree from random lists

“…There are some previous results on coloring graphs from random lists of nonconstant size in the literature: In , it is proved that there is a constant c > 0 such that if L is a random ‐list, assignment for , where is the line graph of the balanced complete bipartite graph on n + n vertices and , then whp there is an L ‐coloring of . Note that in this result is formulated in the language of arrays and Latin squares.…”

“…In this section, we demonstrate how the methods from Sections 2 and 3 can be used for proving some analogous results on list coloring when the size k of the lists in a random (k, C)-list assignment is a (slowly) increasing function of n. We shall derive such analogues of several of the results in the preceding sections. There are some previous results on coloring graphs from random lists of nonconstant size in the literature: In [2], it is proved that there is a constant c > 0 such that if L is a random (k, {1, . .…”

“…, n})-list, assignment for L(K n,n ), where L(K n,n ) is the line graph of the balanced complete bipartite graph on n + n vertices and k > (1 − c)n, then whp there is an L-coloring of L(K n,n ). Note that in [2] this result is formulated in the language of arrays and Latin squares.…”