Graph-Different Permutations

Körner, János; Malvenuto, Claudia; Simonyi, Gábor

doi:10.1137/070692686

Cited by 13 publications

(42 citation statements)

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“…Needless to say, we do not know the answer, as our best upper bound on R( L) for any orientation L of L is just the trivial value 1. From the other side, Proposition 17 gives the best lower bound we know on any single orientation of L. For L itself, the best lower bound published so far is the one in [14] having value 1 4 log 10 ≈ 0.83048. Next we improve on this lower bound.…”

Section: Proof Of Theorem 15mentioning

confidence: 95%

“…This problem was considered by Tuza in [26], where it is solved in the case when p or q is equal to 1. The result in [14] for κ(K 1,r ) translates to this solution. As far as we know, the problem is unsolved for all other pairs of values p and q.…”

Section: Remarkmentioning

confidence: 99%

“…The paper [14] investigated the maximum number of pairwise G-different permutations of [n] for finite graphs G with vertex set [m], m ≤ n. It was observed that, for a fixed finite graph G, this number is constant if n is large enough. This eventual constant value κ(G) was introduced as a new graph invariant: it is straightforward to note that κ(G) does not depend on the actual labelling of the vertices of G by natural numbers.…”

Section: Finite Graphs and Digraphsmentioning

confidence: 99%

“…At the other extreme, the problem of finding κ d for oriented complete graphs, e.g., those of transitive tournaments, is as open as for their undirected counterparts, i.e., the determination of the values κ(K r ), cf. [14]. We do not know even whetherκ(K r ) := lim n→∞ NN(K r ) is superexponential in r.…”

Section: Finite Graphs and Digraphsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Permutation Capacities of Families of Oriented Infinite Paths

Brightwell¹,

Cohen²,

Fachini³

et al. 2010

SIAM J. Discrete Math.

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Körner and Malvenuto asked whether one can find n ⌊n/2⌋ linear orderings (i. e., permutations) of the first n natural numbers such that any pair of them place two consecutive integers somewhere in the same position. This led to the notion of graph-different permutations. We extend this concept to directed graphs, focussing on orientations of the semi-infinite path whose edges connect consecutive natural numbers. Our main result shows that the maximum number of permutations satisfying all the pairwise conditions associated with all of the various orientations of this path is exponentially smaller, for any single orientation, than the maximum number of those permutations which satisfy the corresponding pairwise relationship. This is in sharp contrast with a result of Gargano, Körner, and Vaccaro concerning the analogous notion of Sperner capacity of families of finite graphs. We improve the exponential lower bound for the original problem, and list a number of open questions.

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Section: Proof Of Theorem 15mentioning

confidence: 95%

Section: Remarkmentioning

confidence: 99%

Section: Finite Graphs and Digraphsmentioning

confidence: 99%

Section: Finite Graphs and Digraphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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