Pairwise colliding permutations and the capacity of infinite graphs

Abstract

Abstract. We call two permutations of the first n naturals colliding if they map at least one number to consecutive naturals. We give bounds for the exponential asymptotics of the largest cardinality of any set of pairwise colliding permutations of [n]. We relate this problem to the determination of the Shannon capacity of an infinite graph and initiate the study of analogous problems for infinite graphs with finite chromatic number.

Permutation Capacities of Families of Oriented Infinite Paths

“…This is done along similar lines to those in the proof of Theorem 15 by verifying that the following permutations are pairwise L-different (colliding in the terminology of [13] 1342*** 35214** It is straightforward to check that these 14 permutations are pairwise colliding and thus prove the validity of the recursive lower bound…”

“…The last inequality follows from noting (see [13]) that, for two L-different permutations, the set of positions of odd (even) numbers must differ. (Here we use the notion of being L-different again in the sense of our definitions, identifying L with the symmetrically directed equivalent of its originally undirected version.)…”

“…The natural upper bound N(L, n) ≤ n ⌊n/2⌋ was presented, and conjectured to be sharp, in [13]. It is still an open problem to decide whether N(L, n) is always equal to this upper bound.…”

“…The above definitions naturally extend to digraphs the notion of graph-different permutations investigated in [13,14,17] in the undirected case. In fact, if we identify undirected graphs with their symmetrically directed equivalent, i.e., with digraphs having two oppositely oriented edges in place of all of their undirected edges, then the undirected notion becomes a special case of the directed one.…”

“…The motivating example for introducing graph-different permutations was the puzzle presented in [13] that asks for the value of N(L, n), i.e., the maximum size of a set of permutations of the elements in [n] satisfying that if σ and τ are two distinct permutations in this set, then there is some i ∈ [n] for which |σ(i)−τ (i)| = 1, that is, {σ(i), τ (i)} ∈ E(L). The natural upper bound N(L, n) ≤ n ⌊n/2⌋ was presented, and conjectured to be sharp, in [13].…”