604 research outputs found
Throttling for the game of Cops and Robbers on graphs
We consider the cop-throttling number of a graph for the game of Cops and
Robbers, which is defined to be the minimum of , where
is the number of cops and is the minimum number of
rounds needed for cops to capture the robber on over all possible
games. We provide some tools for bounding the cop-throttling number, including
showing that the positive semidefinite (PSD) throttling number, a variant of
zero forcing throttling, is an upper bound for the cop-throttling number. We
also characterize graphs having low cop-throttling number and investigate how
large the cop-throttling number can be for a given graph. We consider trees,
unicyclic graphs, incidence graphs of finite projective planes (a Meyniel
extremal family of graphs), a family of cop-win graphs with maximum capture
time, grids, and hypercubes. All the upper bounds on the cop-throttling number
we obtain for families of graphs are .Comment: 22 pages, 4 figure
The resolving number of a graph
We study a graph parameter related to resolving sets and metric dimension,
namely the resolving number, introduced by Chartrand, Poisson and Zhang. First,
we establish an important difference between the two parameters: while
computing the metric dimension of an arbitrary graph is known to be NP-hard, we
show that the resolving number can be computed in polynomial time. We then
relate the resolving number to classical graph parameters: diameter, girth,
clique number, order and maximum degree. With these relations in hand, we
characterize the graphs with resolving number 3 extending other studies that
provide characterizations for smaller resolving number.Comment: 13 pages, 3 figure
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
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