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Abstract We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant γ > 0 such that if n is even and A is a 3-dimensional n × n × n array where every cell contains at most γ n symbols, and every symbol occurs at most γ n times in every line of A , then A is avoidable; that is, there is a Latin cube L of order n such that for every i , j , k ∈ { 1 , … , n } , the symbol in position ( i , j , k )...

Abstract A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of S i × S d − i − 1 with 2 d + 2 vertices for all d ≥ 3 and 1 ≤ i ≤ d − 2 ; they also proposed a balanced triangulation of S 1 × S d − 2 with 3 d or 3 d + 2 vertices. In this paper, we provide another centrally symmetric ( 2 d + 2 ) -vertex triangulation of S 2 × S d − 3 . We also construct the fi...

Abstract Let D be a set of positive integers. We study the maximum density μ ( D ) of integral sequences in which the separation between any two terms does not fall in D . The sets D considered in this article are mainly of the form { 1 , j , k } . The closely related function κ ( D ) , the parameter involved in the “lonely runner conjecture,” is also investigated. Exact values of κ ( D ) and μ ( D ) are found for many families of D = { 1 , j , k } . In particular, we prove that the boundary con...

Abstract We study the notion of quasiperiodicity, in the sense of “coverability”, for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger results about quasiperiods of Sturmian words. We demonstrate this by showing that all biinfinite Sturmian words have infinitely many quasiperiods, which is not quite (but almost) true in the right infinite case, and giving a characterization of those qua...

Abstract Given a graph H , a graphic sequence π is potentially H -graphic if there is a realization of π containing H as a subgraph. In 1991, Erdős, Jacobson and Lehel introduced the following problem: determine the minimum even integer σ ( H , n ) such that each n -term graphic sequence with sum at least σ ( H , n ) is potentially H -graphic. This problem can be viewed as a “potential” degree sequence relaxation of the Turan problems. For an arbitrary graph H of order k , Ferrara et al. (2016) ...

Abstract The Mullineux involution is an important map on p -regular partitions that originates from the modular representation theory of S n . In this paper we study the Mullineux transpose map and the generalized column regularization and prove a condition under which the two maps are exactly the same. Our results generalize the work of Bessenrodt, Olsson and Xu, and the combinatorial constructions is related to the Iwahori–Hecke algebra and the global crystal basis of the basic U q ( sl b ) -m...

Abstract Let D be a directed graph. The minimum semidegree of D is defined to be the minimum value of the minimum outdegree and the minimum indegree of D . For nonempty sets S , T ⊆ V ( D ) , we use e ( S , T ) to denote the number of arcs in D from S to T . If D has m arcs and positive minimum semidegree, then we show that D admits a bipartition V ( D ) = V 1 ∪ V 2 such that min { e ( V 1 , V 2 ) , e ( V 2 , V 1 ) } ≥ ( 1 ∕ 6 + o ( 1 ) ) m . We also prove that if the minimum semidegree is at le...

Abstract Let G be a graph. The chromatic edge-stability number es χ ( G ) of a graph G is the minimum number of edges of G whose removal results in a graph H with χ ( H ) = χ ( G ) − 1 . A Nordhaus–Gaddum type inequality for the chromatic edge-stability number is proved. Sharp upper bounds on es χ are given for general graphs in terms of size and of maximum degree, respectively. All bounds are demonstrated to be sharp. Graphs with es χ = 1 are considered and in particular characterized among k -...

Abstract Let S n denote the set of permutations of { 1 , 2 , … , n } . The function f ( n , s ) is defined to be the minimum size of a subset S ⊆ S n with the property that for any ρ ∈ S n there exists some σ ∈ S such that the Hamming distance between ρ and σ is at most n − s . The value of f ( n , 2 ) is the subject of a conjecture by Kezdy and Snevily, which implies several famous conjectures about Latin squares. We prove that the odd n case of the Kezdy–Snevily Conjecture implies the whole co...

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