On Hamilton Cycles in Locally ConnectedGraphs with Vertex Degree Constraints

Yury L. Orlovich 1,2

Faculty of Applied Mathematics and Computer Science,Belarus State University, Minsk, Belarus

Valery S. Gordon 1,3

United Institute of Informatics Problems,National Academy of Sciences of Belarus, Minsk, Belarus

Chris N. Potts 1,4

School of Mathematics,University of Southampton, Southampton, U.K.

Vitaly A. Strusevich 1,5

School of Computing and Mathematical Sciences,University of Greenwich, London, U.K.

Abstract

It is shown that every connected, locally connected graph with the maximum vertexdegree Δ(G) = 5 and the minimum vertex degree δ(G) ≥ 3 is fully cycle extendable.For Δ(G) ≤ 4, all connected, locally connected graphs, including infinite ones, areexplicitly described. The Hamilton Cycle problem for locally connected graphswith Δ(G) ≤ 7 is shown to be NP-complete.

Keywords: Hamiltonian graph, local connectivity, NP-completeness.

Electronic Notes in Discrete Mathematics 29 (2007) 169–173

1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2007.07.028

1 Introduction

In this paper, we study hamiltonian properties of locally connected graphswith a bounded vertex degree. We shall use the terminology of [1].

Let G be a graph with the vertex set V (G) and the edge set E(G). GraphG is hamiltonian if G has a Hamilton cycle, i.e., a cycle containing all verticesof G. A cycle C in a graph G is extendable if there exists a cycle C ′ in G suchthat V (C) ⊂ V (C ′) and |V (C ′)| = |V (C)| + 1. A connected graph G is fullycyclic extendable if every vertex of G is on a triangle and every non-Hamiltoncycle is extendable. Clearly, any fully cycle extendable graph is hamiltonian.Graph G is locally connected if for any vertex u of G the neighborhood of u(i.e., the set of vertices adjacent to u) induces a connected subgraph of G.

As usual, Pn and Cn denote the path and the cycle, On and Kn denotethe empty and complete graphs on n vertices, respectively. Let Wn denotethe wheel on n + 1 vertices, and K1,1,q denote the complete 3-partite graphwith two parts of size 1 and one part of size q. Let P1,∞ denote the one-wayinfinite path, i.e., a graph with V (P1,∞) = {xk | k ∈ N} and E(P1,∞) ={xkxk+1 | k ∈ N}. Similarly, P∞,∞ denote the two-way infinite path withV (P∞,∞) = {xk | k ∈ Z}, E(P∞,∞) = {xkxk+1 | k ∈ Z}.

It is well-known that the problem of deciding whether a given graph ishamiltonian, is NP-complete, and it is natural to look for conditions for theexistence of a Hamilton cycle for special classes of graphs. Study of hamil-tonicity conditions for graphs with a prescribed local structure is one of ratherimportant directions in graph theory. The first results in this area were ob-tained in the 70s and 80s of the last century by Chartrand, Pippert, Gould andPolimeni [2,3], Kikust [6,7], Oberly and Sumner [8]. The basis of the approachdeveloped in these papers is a special graph property that being mapped ontoa local structure of a graph leads to the existence of a Hamilton cycle. Thestudy of Hamilton cycles in locally connected graphs with a bounded vertexdegree was initiated by Kikust [7] and Hendry [5]. Kikust [7] showed that eachconnected, locally connected 5-regular graph is hamiltonian, and Hendry [5]proved that every connected, locally connected graph G with Δ(G) ≤ 5 andΔ(G) − δ(G) ≤ 1 is fully cycle extendible.

In Section 2, for Δ(G) ≤ 4, we explicitly describe all connected, locally

1 Partially supported by INTAS (Project 03-51-5501)2 Email: orlovich@bsu.by3 Email: gordon@newman.bas-net.by4 Email: cnp@maths.soton.ac.uk5 Email: v.strusevich@greenwich.ac.uk

Y.L. Orlovich et al. / Electronic Notes in Discrete Mathematics 29 (2007) 169–173170

connected graphs, including infinite ones. In Section 3, we show that everyconnected, locally connected graph with the maximum vertex degree Δ(G) = 5and the minimum vertex degree δ(G) ≥ 3 is fully cycle extendable which ex-tends the results of Kikust and Hendry on fully cycle extendability of con-nected, locally connected graphs with Δ(G) bounded by 5. In Section 4, theHamilton Cycle problem for locally connected graphs with Δ(G) ≤ 7 isshown to be NP-complete.

2 Locally connected graphs with Δ(G) ≤ 4

One of the first results on finite connected, locally connected graphs withΔ(G) ≤ 4 establishes the following condition of hamiltonicity [2]: such graphsare either hamiltonian or isomorphic to K1,1,3. Hendry [5] shows that con-nected, locally connected graphs with Δ(G) ≤ 5 and Δ(G) − δ(G) ≤ 1 arefully cycle extendable.

All these properties are covered for Δ(G) ≤ 4 by the following theoremthat explicitly describes connected, locally connected graphs with Δ(G) ≤ 4.

Theorem 2.1 Let G be a connected, locally connected (not necessarily finite)graph with Δ(G) ≤ 4. The following claims hold.

If Δ(G) = δ(G), then G ∈ {C2n : n ≥ 3} ∪ {P 2

∞,∞}.If Δ(G) − δ(G) = 1, then G ∈ {P 2

4 , K5 − e, W4, O2 + P4}.If Δ(G) − δ(G) = 2, then G ∈ {K1,1,3, K2 + P3, H1, H2, H3, H4} ∪ {P 2

n :n ≥ 5} ∪ {P 2

1,∞}. The graphs H1, H2, H3, and H4 are shown in Fig. 1.

Fig. 1. Graphs H1, H2, H3, and H4.

3 Cyclic properties of locally connected graphs withΔ(G) ≥ 5

Since Theorem 2.1 explicitly describes all connected, locally connected graphswith Δ(G) ≤ 4, it is interesting to find the hamiltonian properties of thesegraphs under Δ(G) ≥ 5 enhancing the results of Kikust [7] and Hendry [5].

Y.L. Orlovich et al. / Electronic Notes in Discrete Mathematics 29 (2007) 169–173 171

In particular, the following question arises: Is it true that each connected,locally connected graph G with Δ(G) = 5 and δ(G) ≥ 3 is hamiltonian? Thefollowing theorem answers this question.

Theorem 3.1 Let G be a connected, locally connected graph with Δ(G) = 5and δ(G) ≥ 3. Then G is fully cycle extendable.

Note that we cannot use the stronger inequality δ(G) ≥ 2 in Theorem 3.1since graph K1,1,4 with δ = 2 and Δ = 5 is not hamiltonian. Theorem 3.2gives another examples of nonhamiltonian graphs with δ = 2, Δ = 5 andtogether with Theorem 3.1 enhances the results of Kikust [7] and Hendry [5]on hamiltonicity of connected, locally connected graph with Δ(G) = 5.

Theorem 3.2 Let G be a connected, locally connected graph with Δ(G) = 5and do not contain the graph F from Fig. 2 as induced subgraph. Then eitherG is hamiltonian or G ∈ F ∪ {G1, G2, G3}. Here graphs G1, G2, and G3 areshown in Fig. 2 and F is a class of connected, locally connected graphs H withΔ(H) = 5 and with such four vertices u, v, x, y that degH x = degH y = 2 anduv ∈ E(H), ux ∈ E(H), uy ∈ E(H), vx ∈ E(H), vy ∈ E(H) but xy �∈ E(H).

Fig. 2. Graphs F , G1, G2, and G3.

Now we turn to the conditions of hamiltonicity of locally connected graphsunder Δ(G) > 5. Let F(r) be the class of connected, locally connected r-regular graphs G such that r ≥ 6 and each edge of G belongs to at least r− 4triangles. Kikust [6] has shown that any G ∈ F(r) is hamiltonian. Adoptinghis method, the following stronger statement can be proved.

Theorem 3.3 Each graph in class F(r) is fully cycle extendable.

4 Complexity of the Hamilton Cycle problem for lo-cally connected graphs with Δ(G) ≤ 7

Consider the following well-known decision problem.

Y.L. Orlovich et al. / Electronic Notes in Discrete Mathematics 29 (2007) 169–173172

Hamilton Cycle

Instance: A graph G.

Question: Does G contain a Hamilton cycle?

The problem is NP-complete for general graphs and remains difficult forgraphs in many special classes [4]. For locally connected graphs with Δ(G) ≤ 4the Hamilton Cycle problem is solvable in polynomial time, see Theo-rem 2.1. On the other hand, the following theorem shows that the HamiltonCycle problem is hard for locally connected graphs with Δ(G) ≤ 7.

Theorem 4.1 The Hamilton Cycle problem is NP-complete for locallyconnected graphs with Δ(G) ≤ 7.

Let Δ∗ be a maximum integer such that the Hamilton Cycle problemfor locally connected graphs with Δ(G) ≤ Δ∗ is polynomially solvable. As animmediate consequence of Theorems 2.1 and 4.1, we can restrict the range ofΔ∗ to 4 ≤ Δ∗ ≤ 6. Furthermore, we conjecture that Δ∗ = 6.

References

[1] Bondy, J. A., and U. S. R. Murty, “Graph Theory with Applications”,Macmillan, London and Elsevier, NewYork, 1976.

[2] Chartrand, G., and R. Pippert, Locally connected graphs, Casopis Pest. Mat.99 (1974), 158–163.

[3] Chartrand, G., R. J. Gould, and A. D. Polimeni, A note on locally connectedand hamiltonian-connected graphs, Israel J. Math. 33 (1979), 5–8.

[4] Garey, M. R., and D. S. Johnson, “Computers and Intractability: A Guide toThe Theory of NP-Completeness”, W.H. Freeman and Co., San Francisco, 1979.

[5] Hendry, G. R. T., A strengthening of Kikust’s theorem, J. Graph Theory 13(1989), 257–260.

[6] Kikust, P. B., A hamiltonian cycle in a regular graph, A VINITI DepositedManuscript, No 5666-73, 20 March 1973, 36 pp, (in Russian).

[7] Kikust, P. B., On the existence of a hamiltonian cycle in a regular graph ofdegree 5, Latvian Math. Yearbook 16 (1975), 33–38, (in Russian).

[8] Oberly, D. J., and D. P. Sumner, Every connected, locally connected nontrivialgraph with no induced claw is hamiltonian, J. Graph Theory 3 (1979), 351–356.

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