69 research outputs found
Hamilton paths with lasting separation
We determine the asymptotics of the largest cardinality of a set of Hamilton
paths in the complete graph with vertex set [n] under the condition that for
any two of the paths in the family there is a subpath of length k entirely
contained in only one of them and edge{disjoint from the other one
Interlocked permutations
The zero-error capacity of channels with a countably infinite input alphabet
formally generalises Shannon's classical problem about the capacity of discrete
memoryless channels. We solve the problem for three particular channels. Our
results are purely combinatorial and in line with previous work of the third
author about permutation capacity.Comment: 8 page
Skewincidence
We introduce a new class of problems lying halfway between questions about
graph capacity and intersection. We say that two binary sequences x and y of
the same length have a skewincidence if there is a coordinate i for which
x_i=y_{i+1}=1 or vice versa. We give rather sharp bounds on the maximum number
of binary sequences of length n any pair of which has a skewincidence
Local chromatic number and Sperner capacity
We introduce a directed analog of the local chromatic number defined by Erdos et al. [Discrete Math. 59 (1986) 21-34] and show that it provides an upper bound for the Sperner capacity of a directed graph. Applications and variants of this result are presented. In particular, we find a special orientation of an odd cycle and show that it achieves the maximum of Sperner capacity among the differently oriented versions of the cycle. We show that apart from this orientation, for all the others an odd cycle has the same Sperner capacity as a single edge graph. We also show that the (undirected) local chromatic number is bounded from below by the fractional chromatic number while for power graphs the two invariants have the same exponential asymptotics (under the co-normal product on which the definition of Sperner capacity is based). We strengthen our bound on Sperner capacity by introducing a fractional relaxation of our directed variant of the local chromatic number. (C) 2005 Elsevier Inc. All rights reserved
- …