6,772 research outputs found
Lower Bounds for the Cop Number When the Robber is Fast
We consider a variant of the Cops and Robbers game where the robber can move
t edges at a time, and show that in this variant, the cop number of a d-regular
graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on
the order of cages, this implies that the cop number of a connected n-vertex
graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This
improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh
(Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also
conjecture a general upper bound O(n^{t/t+1}) for the cop number in this
variant, generalizing Meyniel's conjecture.Comment: 5 page
A formulation of a (q+1,8)-cage
Let be a prime power. In this note we present a formulation for
obtaining the known -cages which has allowed us to construct small
--graphs for and . Furthermore, we also obtain smaller
-graphs for even prime power .Comment: 14 pages, 2 figure
Computational determination of (3,11) and (4,7) cages
A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a
(k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by
Balaban in 1973 is minimal and unique. We also show that the order of a
(4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on
the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The
methods used were a combination of heuristic hill-climbing and an innovative
backtrack search
A construction of small (q-1)-regular graphs of girth 8
In this note we construct a new infinite family of -regular graphs of
girth and order for all prime powers , which are the
smallest known so far whenever is not a prime power or a prime power plus
one itself.Comment: 8 pages, 2 figure
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