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), @a(G), @k(G) and @s"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{@s"4(G)[email protected](G)[email protected](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{@s"4(G)[email protected](G)[email protected](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)>[email protected]"4(G)[email protected](G)-4. (iii) If G is a 4-connected graph and some longest cycle is not a D"3-cycle, then c(G)>[email protected]"4(G)-8. For each condition, the lower bound of the circumference is sharp.