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 $2^m$-cycles for $m \leq 4$ and for $m \leq 5$.Comment: 7 page

A Lower Bound for R(5,6)

The known lower bound for the the classical Ramsey number $R(5,6)$ is improved from $58$ to $59$. 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

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

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

Bounds for the smallest k-chromatic graphs of given girth

Let n(g)(k) denote the smallest order of a k-chromatic graph of girth at least g. We consider the problem of determining n(g)(k) for small values of k and g. After giving an overview of what is known about n(g)(k), we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for k = 4 and g > 5. In particular, the new bounds include: n(4)(7) <= 77, 26 <= n(6)(4) <= 66 and 30 <= n(7)(4) <= 171