168 research outputs found
Three Graphs and the Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture
Three graphs related to the \EGC\, are presented. The graphs are derived from
the Buckyball, the Petersen graph, and the Tutte-Coxeter graph. The first graph
is a partial answer to a question posed by Heckman and Krakovski \cite{planar}
in their recent work on the planar version of the conjecture. The other two
graphs appear to be the smallest known cubic graphs with no -cycles for and for .Comment: 7 page
A Lower Bound for R(5,6)
The known lower bound for the the classical Ramsey number is
improved from to . The method used to construct the graph is a simple
variant of computational methods that have been previously used to construct
Ramsey graphs. The new method uses the concurrent programming features of the
{\em Go} programming language
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
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
On line disjoint paths of bounded length
AbstractIn a recent paper Lovász, Neumann-Lara and Plummer proved some Mengerian theorems for paths of bounded length. In this note the line connectivity analogue of their problem is considered
On Mixed Cages
Mixed graphs have both directed and undirected edges. A mixed cage is a
regular mixed graph of given girth with minimum possible order. In this paper
mixed cages are studied. Upper bounds are obtained by general construction
methods and computer searches
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