Graph Invariants and Large Cycles : A Survey

Graph invariants provide a powerful analytical tool for investigation of abstract substructures of graphs. This paper is devoted to large cycle substructures, namely, Hamilton, longest and dominating cycles and some generalized cycles including Hamilton and dominating cycles as special cases. In this paper, we have collected 36 pure algebraic relations between basic initial graph invariants ensuring the existence of a certain type of large cycles. These simplest kind of relations having no forerunners in the area actually form a source from which nearly all possible hamiltonian results including well-known Ore’s theorem, Posa’s theorem, and many other generalizations can be developed further by various additional new ideas, generalizations, extensions, restrictions, and structural limitations.


Introduction
Graph invariants provide a powerful and may be the single analytical tool for investigation of abstract structures of graphs.They, combined in convenient algebraic relations, contain global and general information about a graph and its particular substructures such as cycle structures, factors, matchings, colorings, and coverings.The discovery of these relations is the primary problem of graph theory.
This paper is devoted to large cycle substructures, perhaps the most important cycle structures in graphs: Hamilton, longest and dominating cycles and some generalized cycles including Hamilton and dominating cycles as special cases.
In the literature, eight basic initial invariants of a graph G are known having significant impact on large cycle structures, namely, order n, size q, minimum degree δ, connectivity κ, independence number α, toughness τ and the lengths of a longest path and a longest cycle in G \ C for a given longest cycle C, denoted by p and c, respectively.
In this paper we have collected 36 pure algebraic relations between basic graph invariants ensuring the existence of a certain type of large cycles.The majority of these results are sharp in all respects.

International Journal of Mathematics and Mathematical Sciences
Focusing only on basic graph invariants, as well as on pure algebraic relations between these parameters, in fact, we present the simplest kind of relations for large cycles having no forerunners in the area.Actually they form a source from which nearly all possible hamiltonian results including well-known Ore's theorem, Posa's theorem, and many other generalizations can be developed further by various additional new ideas, generalizations, extensions, restrictions, and structural limitations such as: i generalized and extended graph invariants: degree sequences P ósa type, Chvatal type , degree sums Ore type, Fun type , neighborhood unions, generalized degrees, local connectivity, and so on, ii extended list of path and cycle structures: Hamilton, longest and dominating cycles, generalized cycles including Hamilton and dominating cycles as special cases, 2-factor, multiple Hamilton cycles, edge disjoint Hamilton cycles, powers of Hamilton cycles, k-ordered Hamilton cycles, arbitrary cycles, cycle systems, pancyclic-type cycle systems, cycles containing specified sets of vertices or edges, shortest cycles, analogous path structures, and so on, iii structural (descriptive) limitations: regular, planar, bipartite, chordal and interval graphs, graphs with forbidden subgraphs, Boolean graphs, hypercubes, and so on, iv graph extensions: hypergraphs, digraphs and orgraphs, labeled and weighted graphs, infinite graphs, random graphs, and so on.
These 36 initial relations are quite sufficient for interested reader to make a clear imagination about developmental mechanisms in hamiltonian graph theory including the origins, current processes, and future possible developments along with various research problems.
We refer to 1-3 for more background and general surveys.The order n, size q, and minimum degree δ clearly are easy computable graph invariants.In 4 , it was proved that connectivity κ can be determined in polynomial time, as well.Determining the independence number α and toughness τ are shown in 5, 6 to be NP -hard problems.Moreover, it was proved 6 that for any positive rational number t, recognizing t-tough graphs in particular 1-tough graphs is an NP -hard problem.
The order n and size q are neutral with respect to cycle structures.Meanwhile, they become more effective combined together Theorem 3.1 .The minimum degree δ having high frequency of occurrence in different relations is, in a sense, a more essential invariant than the order and size, providing some dispersion of the edges in a graph.The combinations between order n and minimum degree δ become much more fruitful especially under some additional connectivity conditions.The impact of some relations on cycle structures can be strengthened under additional conditions of the type δ ≥ α ± i for appropriate integer i.By many graph theorists, the connectivity κ is at the heart of all path and cycle questions providing comparatively more uniform dispersion of the edges.An alternate connectedness measure is toughness τ-the most powerful and less investigated graph invariant introduced by Chvátal 7 as a means of studying the cycle structure of graphs.Chvátal 7 conjectured that there exists a finite constant t 0 such that every t 0 -tough graph is hamiltonian.This conjecture is still open.We have omitted a number of results involving toughness τ as a parameter since they are far from being best possible.
Large cycle structures are centered around well-known Hamilton spanning cycles.Other types of large cycles were introduced for different situations when the graph contains no Hamilton cycles or it is difficult to find it.Generally, a cycle C in a graph G is a large cycle if it dominates some certain subgraph structures in G in a sense that every such structure has a vertex in common with C. When C dominates all vertices in G then C is a Hamilton cycle.When C dominates all edges in G then C is called a dominating cycle introduced by Nash-Williams 8 .Further, if C dominates all paths in G of length at least some fixed integer λ then C is a PD λ path dominating -cycle introduced by Bondy 9 .Finally, if C dominates all cycles in G of length at least λ then C is a CD λ cycle dominating -cycle, introduced in 10 .The existence problems of generalized PD λ and CD λ -cycles are studied in 10 .
Section 2 is devoted to necessary notation and terminology.In Section 3, we discuss pure relations between various basic invariants of a graph and Hamilton cycles.Next sections are devoted to analogous pure relations concerning dominating cycles Section 4 , CD λcycles Section 5 , long cycles Section 6 , long cycles with Hamilton cycles Section 7 , long cycles with dominating cycles Section 8 , and long cycles with CD λ -cycles Section 9 .

Terminology
Throughout this paper we consider only finite undirected graphs without loops or multiple edges.A good reference for any undefined terms is 11 .We reserve n, q, δ, κ, and α to denote the number of vertices order , number of edges size , minimum degree, connectivity, and independence number of a graph, respectively.Each vertex and edge in a graph can be interpreted as simple cycles of lengths 1 and 2, respectively.A graph G is hamiltonian if G contains a Hamilton cycle, that is, a cycle containing every vertex of G.The length c of a longest cycle in a graph is called the circumference.For C a longest cycle in G, let p and c denote the lengths of a longest path and a longest cycle in In particular, PD 0 -cycles and CD 1 -cycles are well-known Hamilton cycles and PD 1 -cycles and CD 2 -cycles are often called dominating cycles.Let s G denote the number of components of a graph G.A graph G is t-tough if |S| ≥ ts G \ S for every subset S of the vertex set V G with s G \ S > 1.The toughness of G, denoted τ G , is the maximum value of t for which G is t-tough taking τ K n ∞ for all n ≥ 1 .Let a, b, t, k be integers with k ≤ t.We use H a, b, t, k to denote the graph obtained from tK a K t by taking any k vertices in subgraph K t and joining each of them to all vertices of K b .Let L δ be the graph obtained from 3K δ K 1 by taking one vertex in each of three copies of K δ and joining them each to other.For odd n ≥ 15, construct the graph G n from K n−1 /2 K δ K n 1 /2−δ , where n/3 ≤ δ ≤ n − 5 /2, by joining every vertex in K δ to all other vertices and by adding a matching between all vertices in K n 1 /2−δ and n 1 /2 − δ vertices in K n−1 /2 .It is easily seen that G n is 1-tough but not hamiltonian.A variation of the graph G n , with K δ replaced by K δ and δ n − 5 /2, will be denoted by G * n .

Hamilton Cycles
We begin with a size lower bound insuring the existence of a Hamilton cycle based on the idea that if a sufficient number of edges are present in the graph on n vertices, then a Hamilton cycle will exist.
Example for Sharpness.To see that the size bound n 2 −3n 5 /2 in Theorem 3.1 is best possible, note that the graph formed by joining one vertex of K n−1 to K 1 , contains n 2 − 3n 4 /2 edges and is not hamiltonian.