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Cyclic Properties of Locally Connected Graphs with Bounded Vertex Degree

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## Cyclic Properties of Locally Connected Graphs with Bounded Vertex Degree

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**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**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**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.**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.**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.**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**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}.**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**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.**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.**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**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.**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.**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?**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).**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.