108 research outputs found
Partitioning orthogonal polygons into at most 8-vertex pieces, with application to an art gallery theorem
We prove that every simply connected orthogonal polygon of vertices can
be partitioned into (simply
connected) orthogonal polygons of at most 8 vertices. It yields a new and
shorter proof of the theorem of A. Aggarwal that mobile guards are sufficient to control the interior of
an -vertex orthogonal polygon. Moreover, we strengthen this result by
requiring combinatorial guards (visibility is only required at the endpoints of
patrols) and prohibiting intersecting patrols. This yields positive answers to
two questions of O'Rourke. Our result is also a further example of the
"metatheorem" that (orthogonal) art gallery theorems are based on partition
theorems.Comment: 20 pages, 12 figure
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
Generalized Tur\'an problems for even cycles
Given a graph and a set of graphs , let
denote the maximum possible number of copies of in an -free
graph on vertices. We investigate the function , when
and members of are cycles. Let denote the cycle of
length and let . Some of our main
results are the following.
(i) We show that for any .
Moreover, we determine it asymptotically in the following cases: We show that
and that the maximum
possible number of 's in a -free bipartite graph is .
(ii) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds,
then for any we have . We prove that forbidding any other even cycle
decreases the number of 's significantly: For any , we have
More generally,
we show that for any and such that , we have
(iii) We prove provided a
strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true
when ). Moreover, forbidding one more cycle decreases the number
of 's significantly: More precisely, we have and for .
(iv) We also study the maximum number of paths of given length in a
-free graph, and prove asymptotically sharp bounds in some cases.Comment: 37 Pages; Substantially revised, contains several new results.
Mistakes corrected based on the suggestions of a refere
A list version of graph packing
We consider the following generalization of graph packing. Let and be graphs of order and a bipartite graph. A bijection from
onto is a list packing of the triple if implies and for all . We extend the classical results of Sauer and Spencer and Bollob\'{a}s
and Eldridge on packing of graphs with small sizes or maximum degrees to the
setting of list packing. In particular, we extend the well-known
Bollob\'{a}s--Eldridge Theorem, proving that if , and , then either packs or is one of 7 possible
exceptions. Hopefully, the concept of list packing will help to solve some
problems on ordinary graph packing, as the concept of list coloring did for
ordinary coloring.Comment: 10 pages, 4 figure
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