63,726 research outputs found
On the complexity of finding and counting solution-free sets of integers
Given a linear equation , a set of integers is
-free if does not contain any `non-trivial' solutions to
. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers has an -free subset
of a given size, and counting all such -free subsets. We also
raise a number of open problems.Comment: 27 page
On Box-Perfect Graphs
Let be a graph and let be the clique-vertex incidence matrix
of . It is well known that is perfect iff the system , is totally dual integral (TDI). In 1982,
Cameron and Edmonds proposed to call box-perfect if the system
, is box-totally dual
integral (box-TDI), and posed the problem of characterizing such graphs. In
this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity
graphs, and identify several other classes of box-perfect graphs. We also
develop a general and powerful method for establishing box-perfectness
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
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