2,969 research outputs found
Equal Sum Sequences and Imbalance Sets of Tournaments
Reid conjectured that any finite set of non-negative integers is the score
set of some tournament and Yao gave a non-constructive proof of Reid's
conjecture using arithmetic arguments. No constructive proof has been found
since. In this paper, we investigate a related problem, namely, which sets of
integers are imbalance sets of tournaments. We completely solve the tournament
imbalance set problem (TIS) and also estimate the minimal order of a tournament
realizing an imbalance set. Our proofs are constructive and provide a
pseudo-polynomial time algorithm to realize any imbalance set. Along the way,
we generalize the well-known equal sum subsets problem (ESS) to define the
equal sum sequences problem (ESSeq) and show it to be NP-complete. We then
prove that ESSeq reduces to TIS and so, due to the pseudo-polynomial time
complexity, TIS is weakly NP-complete.Comment: Presented at the Retrospective Workshop on Discrete Geometry,
Optimization and Symmetry, 25-29 Nov 2013, The Fields Institute, Toronto,
Canad
Imbalances in directed multigraphs
In a directed multigraph, the imbalance of a vertex is defined as
, where and
denote the outdegree and indegree respectively of . We
characterize imbalances in directed multigraphs and obtain lower and upper
bounds on imbalances in such digraphs. Also, we show the existence of a
directed multigraph with a given imbalance set
List-avoiding orientations
Given a graph with a set of forbidden values at each ,
an -avoiding orientation of is an orientation in which for each vertex . Akbari, Dalirrooyfard, Ehsani, Ozeki, and
Sherkati conjectured that if for each , then has an -avoiding orientation, and they showed that this
statement is true when is replaced by . In this
paper, we take a step toward this conjecture by proving that if for each vertex , then has an
-avoiding orientation. Furthermore, we show that if the maximum degree of
is subexponential in terms of the minimum degree, then this coefficient of
can be increased to . Our main
tool is a new sufficient condition for the existence of an -avoiding
orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi
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