cyclic properties of locally connected graphs with bounded vertex degree n.
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Cyclic Properties of Locally Connected Graphs with Bounded Vertex Degree

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  1. Cyclic Properties of Locally Connected Graphs with BoundedVertex Degree V. Gordon1, Yu. Orlovich2, C. Potts3 , V.Strusevich4 1 United Institute of Informatics Problems of the NASB, Minsk, Belarus; 2 Institute of Mathematics of the NASB, Minsk, Belarus; 3 University of Southampton, School of Mathematics, Southampton, UK 4University of Greenwich, School of Computer and Mathematical Sciences, London, UK

  2. Plan of the talk • Introduction • Locally connected graphs with maximum vertex degree (G) bounded by 4 • Cyclic properties of locally connected graphs with (G)  5 • Complexity of the HAMILTONIAN CYCLE problem for locally connected graphs with (G)  7

  3. Introduction Let Gbe a simple undirected graph with the vertex set V(G) and the edge set E(G). • For a vertex u of G, the neighborhoodN(u)of u is the set of all vertices adjacent to u. • A graph is locally connected if for each vertex u, the subgraph induced by N(u) is connected. • The degreedeguof vertex u in Gis the number of edges incident with u or, equivalently, degu =|N(u)|. The minimum and maximumdegrees of verticesin V(G)are denoted by  (G)and (G), respectively.

  4. Graph G is hamiltonian if Ghas a hamiltonian cycle, i.e. a cycle containing every vertex of G. In 1989, Hendry introduced the following concept. • A cycle C in a graph G is extendable if there exists a cycle C* in Gsuch that V(C) V(C*) and |V(C*)| = |V(C)| + 1. • A connected graph G is fully cyclic extendable if every vertex of G is on a triangle and every nonhamiltonian cycle is extendable. Clearly, any fully cycle extendable graph is hamiltonian.

  5. Previous Results Theorem A (Chartrand, Pippert, 1974).LetGbe a connected, locally connectedgraph with (G)4.Then either Gis hamiltonian or isomorphic to K1,1,3 (complete 3-partite graph with two parts of size 1 and one part of size 3). Theorem B (Kikust, 1975).LetGbe a connected, locally connectedgraph with (G) =  (G)= 5.Then Gis hamiltonian. Theorem C (Hendry, 1989).LetGbe a connected, locally connectedgraph with (G) 5 and (G)   (G) 1.Then Gis fully cyclic extendable.

  6. Locally Connected Graphs with (G)  4 Theorem 1.LetGbe a connected, locally connected(not necessary finite)graph with(G)4. The following claims hold. The following theorem explicitly describes connected, locally connected graphs with(G)4. If(G) =  (G), then If(G)   (G) = 1, then If(G)  (G) = 2,then

  7. If(G) =  (G), then • In Theorem 1, • Cn is the cycle, • Pn is the path, • Kn is the complete graph, • On is the empty graph, • Wn is the wheel on n vertices, • K1,1,q is the complete 3-partite graph with two parts of size 1 and one part of size q,and • Kn e is the graph obtained from Kn by deleting a single edge. If(G)   (G) = 1, then If(G)  (G) = 2,then For a graph G, is the complement to G and G2 is the square of G. P1, is a one-way infinite path,i.e. a graph with V(P1,)={xk| k N}, E(P1,)={xkxk +1 | k N}. P, is a two-way infinite path,i.e. a graphwith V(P,)={xk| k Z}, E(P,)={xkxk +1 | k Z}.

  8. Fig. 1 represents nonstandard graphs H1, H2, H3, and H4 from Theorem 1. H4 H3 H1 H2 Fig. 1. Graphs H1, H2, H3, and H4

  9. Cyclic Properties of Locally Connected Graphs with (G)  5 Connected, locally connected graphs with (G)4are completely described (Theorem 1). It is interesting to find the hamiltonian properties of locally connected graphs under (G)  5 enhancing the results of Kikust (Theorem B) and Hendry (Theorem C). First of all, the following question arises naturally: Is it correct that connected, locally connected graphGwith(G) = 5and (G)  3 is hamiltonian? The following theorem answers this question affirmatively.

  10. Theorem 2.Let G be a connected, locally connected graph with(G) = 5and (G)  3. Then G is fully cyclic extendable. The stronger inequality  (G)  2 cannot be used in Theorem 2 since graph K1,1,4 with  = 2 is not hamiltonian. Graph K1,1,4 The following theorem gives another examples of nonhamiltonian graphs with = 2 and together with Theorem 2 enhances the results of Kikust and Hendry on hamiltonicity of locally connected graph with  = 5.

  11. Theorem 3.Let G be a connected, locally connected graph with (G) = 5 and do not contain graph F from Fig. 2 as induced subgraph. Then eitherGis hamiltonian or G S{G1, G2, G3}. Here graphs G1, G2, and G3are shown in Fig. 2 and S is a class of connected, locally connected graphs H with (H) = 5 and with such four vertices u, v, x, y that deg x = deg y = 2 and uv  E(H), ux  E(H), uy  E(H), vx  E(H), vy  E(H) but xy E(H). G3 F G1 G2 Fig. 2. Graphs F, G1, G2, and G3

  12. y x u v V(H) Class S is the set of all connected, locally connected graphs H with (H) = 5 and with such four vertices u, v, x, y that deg x = deg y = 2 and uv  E(H), ux  E(H), uy  E(H), vx  E(H), vy  E(H) but xy E(H). It is easy to see that none of the graphs in S is hamiltonian.

  13. The extendability of cycles for locally connected k-regular graphs (k 6) are described by the following theorem. Theorem 4.Fork 6, let G be a connected, locally connected k-regular graph and suppose that every edge of G belongs to at least k – 4 triangles. Then Gis fully cyclic extendable. Recall that a graphGis regular of degreek, or simply k-regular, if  (G) = (G) = k for some k 1.

  14. Complexity of the HAMILTONIAN CYCLE Problem for Locally Connected Graphs with (G)  7 Consider the following well-known decision problem. • HAMILTONIAN CYCLE • Instance: A graph G. • Question: Is Ghamiltonian?

  15. HAMILTONIAN CYCLE is NP-complete for general graphs and remains difficult for: • 3-connected cubic (i.e., 3-regular) planar graphs (Garey, Johnson, and Tarjan, 1976); • bipartite planar graphs of maximum degree 3(Akiyama, Nishizeki, and Saito, 1980); • line graphs (Bertossi, 1981); • grid graphs (Itai, Papadimitriou, Szwarcfiter, 1982); • maximal planar graphs (Chvátal, 1985); • chordal bipartite graph (Muller, 1996). • HAMILTONIAN CYCLE is solvable in polynomial time for: • cographs (Corneil, Lerchs, Stewart-Burlingham, 1981); • proper circular arc graphs (Bertossi, 1983); • interval graphs (Keil, 1985); • co-comparability graphs (Deogun, Steiner, 1994).

  16. For locally connected graphs with (G) 4 the HAMILTONIAN CYCLE problem is solvable in polynomial time (Theorem 1). Theorem 5.The HAMILTONIAN CYCLE problem is NP-complete for locally connected graphs with(G) 7. Proof is done by the reduction from the HAMILTONIAN CYCLE problem for cubic planar bipartite graphs. Let * be a maximum integer such that the HAMILTONIAN CYCLE problem for locally connected graphs with a restriction  * is polynomially solvable. As an immediate consequence of Theorems 1 and 5, we can restrict the range of * to 4 *  6. Conjecture. * = 6.