492 research outputs found
Color the cycles
The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013
Speeding up Deciphering by Hypergraph Ordering
The "Gluing Algorithm" of Semaev [Des.\ Codes Cryptogr.\ 49 (2008), 47--60]
--- that finds all solutions of a sparse system of linear equations over the
Galois field --- has average running time where is the total number of
equations, and is the set of all unknowns actively
occurring in the first equations. Our goal here is to minimize the exponent
of in the case where every equation contains at most three unknowns.
%Applying hypergraph-theoretic methods we prove The main result states that if
the total number of unknowns is equal
to , then the best achievable exponent is between and for some
positive constants and $c_2.
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
The Disjoint Domination Game
We introduce and study a Maker-Breaker type game in which the issue is to
create or avoid two disjoint dominating sets in graphs without isolated
vertices. We prove that the maker has a winning strategy on all connected
graphs if the game is started by the breaker. This implies the same in the
biased game also in the maker-start game. It remains open to
characterize the maker-win graphs in the maker-start non-biased game, and to
analyze the biased game for . For a more restricted
variant of the non-biased game we prove that the maker can win on every graph
without isolated vertices.Comment: 18 page
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