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On a Hypercube Coloring Problem
"... Let k (n) denote the minimum number of colors necessary to color the ndimensional hypercube so that no two vertices that are at distance at most k from each other get the same color. In other words, this is the smallest number of binary codes with minimum distance k + 1 that form a partition o ..."
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Let k (n) denote the minimum number of colors necessary to color the ndimensional hypercube so that no two vertices that are at distance at most k from each other get the same color. In other words, this is the smallest number of binary codes with minimum distance k + 1 that form a partition of the ndimensional binary Hamming space. It is shown that 2 (n) n and 3 (n) 2n as n tends to infinity.
On Flow and TensionContinuous Maps
, 2002
"... A cycle of a graph G is a set C ⊆ E(G) so that every vertex of the graph (V(G), C) has even degree. If G, H are graphs, we define a map φ: E(G) → E(H) to be cyclecontinuous if the preimage of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every brid ..."
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A cycle of a graph G is a set C ⊆ E(G) so that every vertex of the graph (V(G), C) has even degree. If G, H are graphs, we define a map φ: E(G) → E(H) to be cyclecontinuous if the preimage of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every bridgeless graph has a cyclecontinuous mapping to the Petersen graph. Jaeger showed that if this conjecture is true, then so is the 5cycledoublecover conjecture and the Fulkerson conjecture. Cycle continuous maps give rise to a natural quasiorder...
Tension continuous maps  their structure and applications
"... We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tensioncontinuous mappings (shortly mappings). Existence of a mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied ..."
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We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tensioncontinuous mappings (shortly mappings). Existence of a mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see [10]). In this paper we study the relationship of the homomorphism and orders. We stress the similarities and the differences in both deterministic and random setting. Particularly, we prove that order is dense and universal and we solve a problem of M. DeVos et al. ([4]).