For let denote the minimal number such that there is a rainbow in every equinumerous -coloring of for every
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture 1 In the couples problem, if the acceptability graph is a tree, then a stable matching exists for any set of preference lists.
The problem is to give a characterization of the pairs whose tensor product is robust.
Keywords: algebraic independence
Beneš Conjecture ★★★
Given a partition of a finite set , stabilizer of , denoted , is the group formed by all permutations of preserving each block in .
In particular, what about the sequence , where is a permutation of ?
A connected simple graph is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as , where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.
Keywords: strong coloring