On the Circuit Diameter Conjecture

On the Circuit-cocircuit Intersection Conjecture

Oxley has conjectured that for k ≥ 4, if a matroidM has a k-element set that is the intersection of a circuit and a cocircuit, thenM has a (k− 2)-element set that is the intersection of a circuit and a cocircuit. In this paper we prove a stronger version of this conjecture for regular matroids. We also show that the stronger result does not hold for binary matroids.

On a Conjecture about the Randić Index and Diameter

The Randić index of a graph G is defined as R(G) = ∑ u∼v (d(u)d(v))− 2 , where d(u) is the degree of vertex u, and the summation goes over all pairs of adjacent vertices u, v. A conjecture of R(G) for connected graph with n vertices is as follows: R(G)−D √2− n+1 2 and R(G) D 1 2 + √ 2−1 n−1 , where D is the diameter of G. In this paper, we prove that this conjecture is true for unicyclic graphs.

Circuit diameter and Klee-Walkup constructions

Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to minimally-dependent subsystems of the presentation of the polyhedron. These can be understood as the set of all possible edge directions, including edges that may ari...

The Erdős-Sós conjecture for spiders of diameter 9

Progress on the Murty-Simon Conjecture on diameter-2 critical graphs: a survey

A graph G is diameter 2-critical if its diameter is two and the deletion of any edgeincreases the diameter. Murty and Simon conjectured that the number of edges in adiameter-2-critical graph G of order n is at most bn/4c and that the extremal graphsare the complete bipartite graphs Kbn/2c,dn/2e. We survey the progress made to dateon this conjecture, concentrating mainly on recen...