1. An Improved Degree Based Condition for Hamiltonian Cycles November 22, 2005 November 22, 2005

2. Contents Introduction –Hamiltonian Path and Cycle –Hamiltonian Graph Motivation Proposed Condition for Hamiltonicity Proof Significance and Conclusion

3. Introduction A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex and a Hamiltonian path is a spanning path. A graph containing a Hamiltonian cycle is said to be Hamiltonian. It is clear that every graph with a Hamiltonian cycle has a Hamiltonian path but the converse is not necessarily true

4. Motivation The Hamiltonian problem is generally considered to be determining conditions under which a graph contains a spanning cycle. Named after Sir William Rowan Hamilton, this problem traces its origins to the 1850s [4]. The problem of finding whether a graph G is Hamiltonian is proved to be NP-complete for general graphs [1]. No easily testable characterization is known for Hamiltonian graphs. Nor there exists any such condition to test whether a graph contains a Hamiltonian path or not. Hence research efforts have been spent for finding the necessary and sufficient conditions for a graph to be Hamiltonian. (see [4] for detail survey)

5. Motivation(contd.) Since the problem is NP-Complete no polynomial time characterization is possible. The problem has been proved to be NP- Complete even for simple special graphs like planar, cubic, 3-connected and with no face having fewer than 5 edges; bipartite graphs, square of a graph or even HP is provided. So there have been efforts on discovering sufficient conditions based upon different graph parameters.

6. Known sufficient conditions If G is a simple graph with n>=3 and minimum degree >=n/2 then G is Hamiltonian. (Dirac) If G is simple and for every pair of non- adjacent vertices u,v d(u)+d(v)>=n then G is Hamiltonian iff G+uv is Hamiltonian. (Ore) If G is simple then G is Hamiltonian iff its closure is Hamiltonian.(Bondy and Chvatal)

7. Known Results(contd) If d(u)+d(v)>=n for every pair of non-adjacent vertices then G is Hamiltonian. (Ore) Let G=(V,E) be a connected graph on n vertices and P be a longest path in G having length k and with n vertices u and v. Then the following statements must hold:

8. Known Results(contd.) (a) Either l(u,v)>1 or P is a HP contained in a HC. (b) If l(u,v)>=3, then d P (u)+ d P (v) <=k-l(u,v)+2 © If l(u,v)=2, then either d P (u)+ d P (v) <=k or P is a HP and there is a HC. (Rahman and Kaykobad)

9. Known Results(Contd) Let G=(V,E) be a connected graph with n vertices such that for all pairs of distinct non-adjacent vertices u,v we have d(u)+d(v)+l(u,v)>=n+1. Then G has a HP. (Rahman and Kaykobad) This theorem also implies Ore’s condition for Hamiltonicity.

10. New results Rahman and Kaykobad [3] introduced distance as the new parameter. Sufficiency condition of Ore[2] results from Rahman and kaykobad when distance of end vertices of a HP is 2[3]. In this paper, we are going to establish that the same condition[3] forces Hamiltonian cycle to be present in the graph excepting for the case where end points of a Hamiltonian path is at a distance greater than 2.

11. Our Proposed Condition Theorem 1.1 (Ore[2]) If for every pair of distinct non- adjacent vertices u and v of G, then G is Hamiltonian. Theorem 1.2 (Rahman and Kaykobad [3]) Let G(V,E) be a connected graph with n vertices such that for all pairs of distinct non-adjacent vertices we have, then G has a Hamiltonian path. Let us assume Now we reformulate the theorem 1.2 in the following way to ensure that graph G is indeed Hamiltonian. Theorem 1.3 Let G(V,E) be a graph without cut vertices and cut edges such that for all pairs of distinct non-adjacent vertices we have, then G is Hamiltonian

12. Proof Before presenting the proof we need to clarify the following terms in the theorem 1.3: Cut Vertex: A vertex whose deletion along with incident edges results in a graph with more components than the original graph. i.e. vertex a in the figure Cut Edge : An edge of a connected graph is called a cut-edge if removing the edge disconnects the graph. i.e. edge (a,b) is a cut edge. b a (1/10)

13. Proof Since hypothesis of Theorem 1.3 is identical to that of Theorem 1.2, existence of a Hamiltonian path in such graphs is ensured according to [3]. We will prove that the same graph contains a Hamiltonian cycle provided end vertices of a Hamiltonian Path is at a distance of at least 3. Since, sufficiency condition of Ore implies that distance of end vertices of a Hamiltonain Path is two[3], let us consider a Hamiltonian Path H(u,v) for which. We will first show that cannot be greater than or equal to 4 (2/9)

14. Proof v1v1 u1u1 vl u Fig. 1. A possible simple graph jk If (u,u 1 ), (v,v 1 ) exists such that l(u,v)>3 through u 1, v 1, then l(u1,v1)=l(u,v)-2; and number of vertices between u1, v1 is l(u1,v1)-1, which implies to be l(u,v)-3. So, But to satisfy the sufficient condition of Hamiltonicity, we have So we conclude that and u and v is connected to every vertices except l(u,v)-3 vertices between u1, v1 (3/9)

15. Proof To avoid cut vertices, we need to add more edges. If we add an edge (j,k), 1<j<u 1 and v 1 <k<v, then l(u,v) reduces to 3 So we may add edges like (j,l), (m,k). In that case also l(u,v) is reduced (see fig) and that contradicts our assumption of having l(u,v)>3. u Fig. 2. A possible simple graph lu1u1 v1v1 v m jk (4/9)

16. Proof We may argue to have a graph like Fig3. Let us assume that we have such a graph which also satisfies our condition of Hamiltonicity In this case, But d(u) + d(v) + l(u,v) cannot be greater than n+1. otherwise it is a contradiction to our assumption. So, no such graph exists with l(u,v)>3 u l k vm Fig. 3. A possible simple graph with l(u,v)=4 jn (5/9)

17. uv w+1 w Proof (6/9) So we will proceed with and prove that there is a Hamiltonian cycle. Again implies that no vertex can be adjacent to both u and v at the same time. since then would have been a path u-w-v of length 2. Now those u,v are connected to different vertices and they can be connected to at most n-2 vertices. But then implies That is, each of the internal vertices must be connected to exactly one of u or v.

18. Proof Let us assume for clarity of arguments that u is denoted by 1 and v is denoted by n, and all vertices along Hamiltonian Path H(u,v) are denoted by 2, 3… n-1. Since existence of cross over edges(Fig.4) as depicted in the above (Fig. 4) picture implies existence of a Hamiltonian Cycle, we assume that there is no cross-over edges. Fig. 4. Existence of crossover edge (1, i) and (j, n) where j=i+1 n-1 3 2 ni j 1 (6/9)

19. Proof As the graph does not contain cross over edges, we assume that k is the highest index of node which is adjacent to node 1 (Fig 5). In that case all the vertices with index i<k are adjacent to 1 and vice versa for node n. Here the edge (k, k+1) becomes a cut edge. So to avoid such an edge, we need to add more edges. In that case two situations arise (see next slides): nk+1k 1 Fig. 5. A possible simple graph without crossover edges between vertices 1 and n (7/9)

20. Proof Case 1: To avoid cut edge, there exists an edge (i, j) such that 1<i<k and k+1<j<n (Fig 6). In this case we have a Hamiltonian Cycle Here, denotes jump and denotes increasing or decreasing natural sequence i j nk+1k 1 Fig. 6. A simple graph without crossover edges between the non-adjacent vertices 1and n may contain an edge (i,j) where i k+1 (8/9)

21. Proof Case 2: We may have two edges like (i, k+1) and (k, j) where 1<i<k and k+1<j<n (Fig 7) In this case we have a Hamiltonian cycle i i+1 j-1 k+1 j nk 1 Fig. 7. A simple graph without crossover edges between the non-adjacent vertices 1and n may contain edges like (k, j), (k+1,i) where i k+1 (9/9)

22. Conclusion Hence, our proposed condition, which followed from Rahman and Kaykobad[3], ensures Hamiltonian Cycle in the graph. The supremacy of this condition is that this requires less number of edges than similar existing degree related conditions[2] to ensure Hamiltonicity in a graph. The novelty of this approach is the incorporation of distance parameter l(u,v).

23. References [1] Garey M.R. and Johnson D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979. [2] Ore O., Note on Hamiltonian circuits, Amer. Math. Monthly 67(196) 55. [3] Rahman M. Sohel and Kaykobad M., On Hamiltonian cycles and Hamiltonian paths, Information processing Letters 94(2005), 37-51. [4] Gould R. J., Advances on the Hamiltonian Problem - A Survey, Graphs and Combinatorics, Volume 19, Number 1, March 2003, 7-52.