Degree sum conditions for the circumference of 4-connected graphs

We denote the order, the independence number, the connectivity and the minimum degree sum of independent four vertices of a graph G by n ( G ) , α ( G ) , κ ( G ) and σ 4 ( G ) , respectively. The circumference of a graph G , denoted by c ( G ) , is the length of a longest cycle in G . We call a cycle C of a graph G a D k -cycle if the order of each component of G − V ( C ) is at most k − 1 . Our goal is to accomplish the proof of the statement that if G is a 4 -connected graph, then c ( G ) ≥ min { σ 4 ( G ) − κ ( G ) − α ( G ) + 1 , n ( G ) } . In order to prove this, we consider three conditions for the construction of the outside of a longest cycle: (i) If G is a 3 -connected graph and every longest cycle of G is a D 2 -cycle, then c ( G ) ≥ min { σ 4 ( G ) − κ ( G ) − α ( G ) + 1 , n ( G ) } . (ii) If G is a 3 -connected graph and every longest cycle is a D 3 -cycle and some longest cycle is not a D 2 -cycle, then c ( G ) ≥ σ 4 ( G ) − κ ( G ) − 4 . (iii) If G is a 4 -connected graph and some longest cycle is not a D 3 -cycle, then c ( G ) ≥ σ 4 ( G ) − 8 . For each condition, the lower bound of the circumference is sharp.