A Chain Theorem for 3+-Connected Graphs

A New Characterization of P4-connected Graphs

A graph is said to be P 4-connected if for every partition of its vertex-set into two nonempty disjoint sets, some P 4 in the graph has vertices from both sets in the partition. A P 4-chain is a sequence of vertices such that every four consecutive ones induce a P 4. The main result of this work states that a graph is P 4-connected if and only if each pair of vertices is connected by a P 4-chai...

A non-planar version of Tutte's wheels theorem

Thtte's Wheels Theorem states that a minimally 3-connected non-wheel graph G with at least four vertices contains at least one edge e such that the contraction of e from G produces a graph which is both 3-connected and simple. The edge e is said to be non-essential. We show that a minimally 3-connected graph which is non-planar contains at least six non-essential edges. The wheel graphs are the...

A Degree Condition For Graphs to Have Connected (g, f)-Factors

Every longest circuit of a 3-connected, K3, 3-minor free graph has a chord

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no K3,3 minor, and consequently for planar graphs. Carsten Thomassen made the following conjecture [1, 7]: Conjecture 1 (Thomassen) Every longest circuit of a 3-connected graph has a chord. That conjecture has been proved for planar graphs with minimum degree at...

2-walks in 3-connected Planar Graphs

In this we prove that every 3-connected planar graph has closed walk each vertex, none more than twice, such that any vertex visited twice is in a vertex cut of size 3. This both Tutte's Theorem that 4-connected planar graphs are Hamiltonian and the result of Gao and Richter that 3-connected planar graphs have a closed walk visiting each vertex at least once but at most twice.