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Chords of Longest Cycles in Cubic Graphs
Journal of Combinatorial Theory, Series B TB1776 journal of combinatorial theory, Series B 71, 211214 (1997) article no. TB971776
Chords of Longest Cycles in Cubic Graphs Carsten Thomassen Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark Received September 9, 1996
We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and a recent conjecture on a second Hamiltonian cycle by Triesch, Nolles, and Vygen. 1997 Academic Press
1. INTRODUCTION In 1976 I conjectured that every longest cycle in every 3-connected graph has a chord [3, Conjecture 6.11], see also [8, Conjecture 6; 2, Conjecture 8.1; and 11, Conjecture 8.3.14]. In this paper the conjecture is verified for cubic graphs. The proof follows from a general sufficient condition for a second Hamiltonian cycle (using Thomason's lollipop method) combined with the result of Fleischner and Stiebitz [5] (proved with the aid of the Alon and Tarsi result [1]) that every ``cycle plus triangles graph'' is 3-colorable. We prove that every ``cycle plus triangles graph'' has a second Hamiltonian cycle. This verifies a special case of the conjecture made in 1975 by Sheehan [6] that every 4-regular Hamiltonian graph has another Hamiltonian cycle. Motivated by a problem on optimum tours in the Travelling Salesman Problem, Triesch et al. [10] made the following conjecture: Let C be a cycle with n vertices. Let c be a coloring of V(C) in m colors where 2m+1 n. Consecutive vertices are allowed to have the same color. Add all edges between vertices of the same color. Then the resulting graph G has a Hamiltonian cycle distinct from C. We verify this when c is proper (i.e., no two consecutive vertices of C have the same color) and all color classes have at least 3 vertices. The graph of Fleischner [4, Fig. 6] shows that the conjecture fails in general even if c is proper and all color classes have at least 2 vertices. 211 0095-895697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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2. A SECOND HAMILTONIAN CYCLE Lemma 2.1. Let G be a graph with a Hamiltonian cycle C. Suppose that for some set A of vertices, the subgraph G&A has |A| components each of which is a path whose ends are of odd degree in G. Then (1) for every Hamiltonian cycle C$ of G, C$&A=C&A, and (2) each edge of G incident to a vertex of A is included in an even number of Hamiltonian cycles of G. Proof. (1) is obvious. Let e=xy where x # A. Every Hamiltonian path in G which starts with x and e (if any) must end at an end of one of the paths in G&A. That vertex has odd degree in G. It follows by Thomason's lollipop argument [7] (see also [9, Theorem 2.1]) that G has an even number of Hamiltonian cycles containing e. K Theorem 2.2. Let G be a graph with a Hamiltonian cycle C. Let A be a vertex set in G such that (i) A is independent in C (i.e., A contains no two consecutive vertices of C ), and (ii) A is dominating in G&E(C) (i.e., every vertex of G&A is joined to a vertex in A by some chord of C). Then G has a Hamiltonian cycle C$ distinct from C. Moreover, C$ can be chosen such that (iii)
C$&A=C&A, and
(iv) there is a vertex v in A such that one of the two edges of C$ incident with v is in C and the other is not in C. Proof. We consider the following subgraph G$ of G : G$ contains C and, for each vertex x which is adjacent on C to a vertex of A, we let G$ contain precisely one chord of C from x to a vertex of A. So in G$, x has degree 3. Every vertex of C which is not in A and not consecutive to a vertex of A has degree 2 in G$. Note that G$&A has precisely |A| components, each of which is a path. By Lemma 2.1, G$ has a Hamiltonian cycle C$ distinct from C. Clearly, C$ satisfies (iii). To prove (iv) we consider a Hamiltonian cycle C$ of G$ such that C${C and C$ contains as many edges of C as possible. We claim that C$ satisfies (iv). Suppose (reductio ad absurdum) that C$ does not satisfy (iv). Let H be the graph C _ C$. Let A$ be those vertices of A which contain an edge (and hence two edges) in E(C$)"E(C). Then each vertex of A$ has degree 4 in H, each vertex in V(G)"A$ which is adjacent (on C) to a vertex in A$ has degree 3 in H, and all other vertices have degree 2 in H. Also, H&A$
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has precisely |A$| components, each of which is a path whose ends are of odd degree in H. Now let u be a vertex of A$ and let uy be an edge of C$. By Lemma 2.1, H has a Hamiltonian cycle C" distinct from C$ containing uy. There must be some vertex z in A$ such that C" does not contain the two edges of C$ incident with z. (This follows because C"&A$=C$&A$.) But then C" contains an edge of C incident with z. We claim that C" also contains each edge f in C & C$. This follows from Lemma 2.1 if f is not incident with a vertex of A. On the other hand, if f is incident with a vertex w of A, then w is not in A$ (by the definition of A$) and hence w has degree 2 in H. Therefore f is in every Hamiltonian cycle of H. This contradiction to the maximality property of C$ proves that C$ satisfies (iv). K
3. APPLICATIONS The following was proved in [9]. Theorem 3.1. If C is a Hamiltonian cycle in a bipartite graph with bipartition A, B such that every vertex of B has degree at least 3 in G, then G has a Hamiltonian cycle distinct from C. Theorem 3.1 follows from Theorem 2.2 since the set A in the former can play the role of A in the latter. Conversely, Theorem 2.2 except (iv) follows easily from Theorem 3.1 by subdividing each edge of C&A once and adding the new vertices of degree 2 to A. The next result is related to the conjecture of Triesch et al. mentioned in the Introduction. Theorem 3.2. Let G be a graph with a Hamiltonian cycle C such that G&E(C) is the disjoint union of complete graphs each of order at least 3. Then G has a Hamiltonian cycle distinct from C. Proof. Select a triangle in each component of G&E(C). Let G$ denote the union of C and these triangles. Then G$ is a subdivision of a ``cycle plus triangles graph'' which, by a result of Fleischner and Stiebitz [5] is 3-colorable. Since a subdivision of a 3-colorable graph is 3-colorable, G$ is 3-colorable. Any color class of G$ satisfies (in G) the assumption of Theorem 2.2. Hence G has a Hamiltonian cycle distinct from C. K We can now prove the main result. Theorem 3.3. Let C be any longest cycle in a 3-connected cubic graph G. Then C has a chord.
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Proof. Suppose (reductio ad absurdum) that C has no chord. For each component H of G&V(C) we select three vertices x(H), y(H), z(H) of C which are joined to vertices of H. This is possible because G is 3-connected. As G is cubic, [x(H), y(H), z(H)] & [x(H$), y(H$), z(H$)]=< when H{H$. We form a new graph G$ which consists of C and all triangles of the form x(H) y(H) z(H) x(H). By [5], G$ is 3-colorable. Let A be a color class. Assume without loss of generality that A contains all vertices of the form x(H) where H is a component of G&V(C). We form a new graph G$ by contracting each component H of G&V(C) into x(H). Clearly, G$ and A satisfy (i) and (ii) in Theorem 2.2 (with G$ instead of G). Let C$ be a Hamiltonian cycle of G$ satisfying (iii) and (iv). C$ can easily be modified into a cycle C" of G. The condition (iv) implies that C" is longer than C, a contradiction which completes the proof. K
REFERENCES 1. N. Alon and M. Tarsi, Colourings and orientations of graphs, Combinatorica 12 (1992), 125134. 2. B. Alspach and C. D. Godsil (Eds.), Cycles in Graphs, Ann. Discrete Math. 27 (1985). 3. J. A. Bondy, Basic graph theory, in ``Handbook of Combinatorics'' (M. Grotschel, L. Lovasz, and R. L. Graham, Eds.), pp. 3110, North-Holland, Amsterdam, 1995. 4. H. Fleischner, Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs, J. Graph Theory 18 (1994), 449459. 5. H. Fleischner and M. Stiebitz, A solution to a colouring problem, Discrete Math. 101 (1992), 3948. 6. J. Sheehan, The multiplicity of Hamiltonian circuits in a graph, in ``Recent Advances in Graph Theory'' (M. Fiedler, Ed.), pp. 447480, Academia, Prague, 1975. 7. A. G. Thomason, Hamiltonian cycles and uniquely edge-colorable graphs, in ``Advances in Graph Theory'' (B. Bollobas, Ed.); Ann. Discrete Math. 3 (1978), 259259. 8. C. Thomassen, Configurations in graphs of large minimum degree, connectivity or chromatic number, in ``Proc. 3rd International Conference on Combinatorial Mathematics, New York, 1985''; Ann. New York Acad. Sci. 555 (1989), 402412. 9. C. Thomassen, On the number of Hamiltonian cycles in bipartite graphs, Combin. Probab. Comput. 5 (1996), 437442. 10. E. Triesch, W. Nolles, and J. Vygen, ``Die Einsatzplannung von Zementmischern und ein Travelling Salesman Problem,'' manuscript. 11. H.-J. Voss, ``Cycles and Bridges in Graphs,'' Kluwer AcademicDeutcher Verlag der Wissenschaften, DordrechtBerlin, 1991.
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Chords of Longest Cycles in Cubic Graphs Carsten Thomassen Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark Received September 9, 1996
We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and a recent conjecture on a second Hamiltonian cycle by Triesch, Nolles, and Vygen. 1997 Academic Press
1. INTRODUCTION In 1976 I conjectured that every longest cycle in every 3-connected graph has a chord [3, Conjecture 6.11], see also [8, Conjecture 6; 2, Conjecture 8.1; and 11, Conjecture 8.3.14]. In this paper the conjecture is verified for cubic graphs. The proof follows from a general sufficient condition for a second Hamiltonian cycle (using Thomason's lollipop method) combined with the result of Fleischner and Stiebitz [5] (proved with the aid of the Alon and Tarsi result [1]) that every ``cycle plus triangles graph'' is 3-colorable. We prove that every ``cycle plus triangles graph'' has a second Hamiltonian cycle. This verifies a special case of the conjecture made in 1975 by Sheehan [6] that every 4-regular Hamiltonian graph has another Hamiltonian cycle. Motivated by a problem on optimum tours in the Travelling Salesman Problem, Triesch et al. [10] made the following conjecture: Let C be a cycle with n vertices. Let c be a coloring of V(C) in m colors where 2m+1 n. Consecutive vertices are allowed to have the same color. Add all edges between vertices of the same color. Then the resulting graph G has a Hamiltonian cycle distinct from C. We verify this when c is proper (i.e., no two consecutive vertices of C have the same color) and all color classes have at least 3 vertices. The graph of Fleischner [4, Fig. 6] shows that the conjecture fails in general even if c is proper and all color classes have at least 2 vertices. 211 0095-895697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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2. A SECOND HAMILTONIAN CYCLE Lemma 2.1. Let G be a graph with a Hamiltonian cycle C. Suppose that for some set A of vertices, the subgraph G&A has |A| components each of which is a path whose ends are of odd degree in G. Then (1) for every Hamiltonian cycle C$ of G, C$&A=C&A, and (2) each edge of G incident to a vertex of A is included in an even number of Hamiltonian cycles of G. Proof. (1) is obvious. Let e=xy where x # A. Every Hamiltonian path in G which starts with x and e (if any) must end at an end of one of the paths in G&A. That vertex has odd degree in G. It follows by Thomason's lollipop argument [7] (see also [9, Theorem 2.1]) that G has an even number of Hamiltonian cycles containing e. K Theorem 2.2. Let G be a graph with a Hamiltonian cycle C. Let A be a vertex set in G such that (i) A is independent in C (i.e., A contains no two consecutive vertices of C ), and (ii) A is dominating in G&E(C) (i.e., every vertex of G&A is joined to a vertex in A by some chord of C). Then G has a Hamiltonian cycle C$ distinct from C. Moreover, C$ can be chosen such that (iii)
C$&A=C&A, and
(iv) there is a vertex v in A such that one of the two edges of C$ incident with v is in C and the other is not in C. Proof. We consider the following subgraph G$ of G : G$ contains C and, for each vertex x which is adjacent on C to a vertex of A, we let G$ contain precisely one chord of C from x to a vertex of A. So in G$, x has degree 3. Every vertex of C which is not in A and not consecutive to a vertex of A has degree 2 in G$. Note that G$&A has precisely |A| components, each of which is a path. By Lemma 2.1, G$ has a Hamiltonian cycle C$ distinct from C. Clearly, C$ satisfies (iii). To prove (iv) we consider a Hamiltonian cycle C$ of G$ such that C${C and C$ contains as many edges of C as possible. We claim that C$ satisfies (iv). Suppose (reductio ad absurdum) that C$ does not satisfy (iv). Let H be the graph C _ C$. Let A$ be those vertices of A which contain an edge (and hence two edges) in E(C$)"E(C). Then each vertex of A$ has degree 4 in H, each vertex in V(G)"A$ which is adjacent (on C) to a vertex in A$ has degree 3 in H, and all other vertices have degree 2 in H. Also, H&A$
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has precisely |A$| components, each of which is a path whose ends are of odd degree in H. Now let u be a vertex of A$ and let uy be an edge of C$. By Lemma 2.1, H has a Hamiltonian cycle C" distinct from C$ containing uy. There must be some vertex z in A$ such that C" does not contain the two edges of C$ incident with z. (This follows because C"&A$=C$&A$.) But then C" contains an edge of C incident with z. We claim that C" also contains each edge f in C & C$. This follows from Lemma 2.1 if f is not incident with a vertex of A. On the other hand, if f is incident with a vertex w of A, then w is not in A$ (by the definition of A$) and hence w has degree 2 in H. Therefore f is in every Hamiltonian cycle of H. This contradiction to the maximality property of C$ proves that C$ satisfies (iv). K
3. APPLICATIONS The following was proved in [9]. Theorem 3.1. If C is a Hamiltonian cycle in a bipartite graph with bipartition A, B such that every vertex of B has degree at least 3 in G, then G has a Hamiltonian cycle distinct from C. Theorem 3.1 follows from Theorem 2.2 since the set A in the former can play the role of A in the latter. Conversely, Theorem 2.2 except (iv) follows easily from Theorem 3.1 by subdividing each edge of C&A once and adding the new vertices of degree 2 to A. The next result is related to the conjecture of Triesch et al. mentioned in the Introduction. Theorem 3.2. Let G be a graph with a Hamiltonian cycle C such that G&E(C) is the disjoint union of complete graphs each of order at least 3. Then G has a Hamiltonian cycle distinct from C. Proof. Select a triangle in each component of G&E(C). Let G$ denote the union of C and these triangles. Then G$ is a subdivision of a ``cycle plus triangles graph'' which, by a result of Fleischner and Stiebitz [5] is 3-colorable. Since a subdivision of a 3-colorable graph is 3-colorable, G$ is 3-colorable. Any color class of G$ satisfies (in G) the assumption of Theorem 2.2. Hence G has a Hamiltonian cycle distinct from C. K We can now prove the main result. Theorem 3.3. Let C be any longest cycle in a 3-connected cubic graph G. Then C has a chord.
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Proof. Suppose (reductio ad absurdum) that C has no chord. For each component H of G&V(C) we select three vertices x(H), y(H), z(H) of C which are joined to vertices of H. This is possible because G is 3-connected. As G is cubic, [x(H), y(H), z(H)] & [x(H$), y(H$), z(H$)]=< when H{H$. We form a new graph G$ which consists of C and all triangles of the form x(H) y(H) z(H) x(H). By [5], G$ is 3-colorable. Let A be a color class. Assume without loss of generality that A contains all vertices of the form x(H) where H is a component of G&V(C). We form a new graph G$ by contracting each component H of G&V(C) into x(H). Clearly, G$ and A satisfy (i) and (ii) in Theorem 2.2 (with G$ instead of G). Let C$ be a Hamiltonian cycle of G$ satisfying (iii) and (iv). C$ can easily be modified into a cycle C" of G. The condition (iv) implies that C" is longer than C, a contradiction which completes the proof. K
REFERENCES 1. N. Alon and M. Tarsi, Colourings and orientations of graphs, Combinatorica 12 (1992), 125134. 2. B. Alspach and C. D. Godsil (Eds.), Cycles in Graphs, Ann. Discrete Math. 27 (1985). 3. J. A. Bondy, Basic graph theory, in ``Handbook of Combinatorics'' (M. Grotschel, L. Lovasz, and R. L. Graham, Eds.), pp. 3110, North-Holland, Amsterdam, 1995. 4. H. Fleischner, Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs, J. Graph Theory 18 (1994), 449459. 5. H. Fleischner and M. Stiebitz, A solution to a colouring problem, Discrete Math. 101 (1992), 3948. 6. J. Sheehan, The multiplicity of Hamiltonian circuits in a graph, in ``Recent Advances in Graph Theory'' (M. Fiedler, Ed.), pp. 447480, Academia, Prague, 1975. 7. A. G. Thomason, Hamiltonian cycles and uniquely edge-colorable graphs, in ``Advances in Graph Theory'' (B. Bollobas, Ed.); Ann. Discrete Math. 3 (1978), 259259. 8. C. Thomassen, Configurations in graphs of large minimum degree, connectivity or chromatic number, in ``Proc. 3rd International Conference on Combinatorial Mathematics, New York, 1985''; Ann. New York Acad. Sci. 555 (1989), 402412. 9. C. Thomassen, On the number of Hamiltonian cycles in bipartite graphs, Combin. Probab. Comput. 5 (1996), 437442. 10. E. Triesch, W. Nolles, and J. Vygen, ``Die Einsatzplannung von Zementmischern und ein Travelling Salesman Problem,'' manuscript. 11. H.-J. Voss, ``Cycles and Bridges in Graphs,'' Kluwer AcademicDeutcher Verlag der Wissenschaften, DordrechtBerlin, 1991.
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