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Total Z-transformation graphs of perfect matching of plane bipartite graphs
Total Z-transformation graphs of perfect matchings of plane bipartite graphs Heping Zhang,
a;1
Fuji Zhang
b
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China, e-mail address: [email protected] b Department of Mathematics, Xiamen University, Xiamen, Fujian 361005, P. R. China, e-mail address: [email protected]
a
Abstract The total Z-transformation graph and digraph of perfect matchings of a plane bipartite graph are de ned. For a plane elementary bipartite graph it is shown that its total Z-transformation graph and digraph are 2-connected and strongly connected respectively. As an immediate consequence, we have that a plane bipartite graph is elementary if and only if its total Z-transformation digraph is strongly connected. Key words: Plane bipartite graph; Perfect matching; Z-transformation graph
1 Introduction
Let G be a graph with vertex-set V (G) and edge-set E (G). A subset E of E (G) is called a matching of G if no two edges of E share an end-vertex. Further a matching of G is said to be perfect if it covers all vertices of G. An edge of G is said to be allowed if it is contained in a perfect matching of G and forbidden otherwise. Then G is said to be elementary if all allowed edges of G form a connected spanning subgraph of G. It is known that a bipartite graph G is elementary if and only if G is connected and each edge of G is allowed; An elementary bipartite graph with larger than 2 vertices is 2-connected. 0
0
Plane bipartite graphs form an interesting class that contains a family of typical graphs, hexagonal systems, often arising in some real problems. Hexagonal systems with perfect matchings are of chemical signi cance and their topological properties have been extensively studied [1{3,11]. In connection with the 1
supported in part by the national natural science foundation of China (19701014)
Preprint submitted to Elsevier Preprint
7 May 2000
concept of the aromatic sextet for benzenoid hydrocarbons [1], Zhang et al. [4,5] introduced the concept of Z-transformation graphs of perfect matchings of hexagonal systems. By virtue of this concept a complete characterization [6] for hexagonal systems with forcing edges (i.e., those edges belonging to exactly one perfect matching) was given. The present authors in [8] treated general plane bipartite graphs in a uni ed way and also extended naturally the Z-transformation graph of hexagonal systems to general plane bipartite graphs, further in [10] introduced Z-transformation digraph. In mathematical point of view, a more natural de nition for the total Ztransformation (di)graph of a plane bipartite graph involving all the faces is given. In this paper we show that the total Z-transformation graph of a plane elementary bipartite graph is 2-connected; a plane bipartite graph is elementary if and only if the total Z-transformation digraph is strongly connected.
2 Z-transformation graph Let G be a plane bipartite graph with a perfect matching M . A cycle C of G is called an M -alternating cycle if the edges of C appear alternately in M and E (G)nM . An M -alternating path can be de ned similarly. For convenience all the vertices of G are always colored properly black and white; that is, adjacent vertices receive di erent colors. The boundary of an interior face ( nite face) of G is called a ring if it is a cycle. The symmetric di erence of two nite sets A and B is de ned as A B =: (A [ B )n(A \ B ): In the following we restrict our consideration to plane bipartite graphs G with a perfect matching M .
De nition 1 [9]. An M -alternating cycle of G is called a proper M -alternating cycle C if each edge of C belonging to M goes from the white end-vertex to the black end-vertex by the clockwise orientation of C ; otherwise C is known as an improper M -alternating cycle.
De nition 2 Let G be a plane bipartite graph with perfect matchings. The
Z-transformation graph of G, denoted by Z (G), is de ned as a simple graph in which the vertices are the perfect matchings of G and vertices M1 and M2 are adjacent provided the symmetric di erence M1 M2 of the corresponding perfect matchings consists exactly of a ring of G; the Z-transformation digraph, denoted by Z~ (G), is the orientation of Z (G): an edge M1M2 of Z (G) is oriented from M1 to M2 if and only if M1 M2 is a proper M1-alternating ring of G.
For hexagonal systems H the connectivity of Z (H ) is equal to the minimum degree [5]; for polyomino graphs P (or square systems), Z (P ) possesses the same result with the two exceptions [7]. For general plane elementary bipartite 2
graphs G, Z(G) are connected bipartite graphs that may contain cut-vertex [8]. Further, by discussing Z-transformation digraph we obtained the following result.
Theorem 1 [10]. Let G be a plane elementary bipartite graph. Then the blockgraph of the Z-transformation graph of G is a path.
Theorem 2 [10]. Let G be a plane elementary bipartite graph. Then Z~ (G)
contains no directed cycles and has a unique sink t and a unique source s, which belong separately to the two extremal blocks. For any other vertex w, Z~ (G) has a directed path from w to t and a directed path from s to w.
3 Total Z-transformation graph Let G be a plane bipartite graph and C a cycle that is the boundary of a face f of G. The clockwise orientation of C with respect to f is de ned as follows: when going along the direction of cycle C the face f lies on the right side.
De nition 3 Let a plane bipartite graph G have a perfect matching M and
an M -alternating cycle C be the boundary of a face f of G. Then C is called a proper M -alternating boundary if each edge of C belonging to M goes from the white end-vertex to the black end-vertex along the clockwise orientation of C with respect to f ; otherwise C is known as an improper M -alternating boundary.
De nition 4 Let G be a plane bipartite graph with perfect matchings. The
total Z-transformation graph of G, denoted by Z (G), is de ned as a graph in which the vertices are the perfect matchings of G and two perfect matchings M1 and M2 are joined by an edge provided M1 M2 forms a cycle that is the boundary of a face of G; The total Z-transformation digraph of G, denoted Z~ (G), is an orientation of Z (G) such that each edge M1M2 is oriented from M1 to M2 if and only if M1 M2 is a proper M1-alternating boundary of G. t
t
t
Theorem 3 Let G be a plane elementary bipartite graph with more than one cycle. Then the total Z-transformation graph Z (G) is 2-connected. t
Theorem 4 Let a 2-connected plane bipartite graph G have perfect matchings
and do have more than one cycle. Then either Z (G) is 2-connected or each component of Z (G) is 2-connected. t
t
Lemma 5 Let G be a plane elementary bipartite graph. Let M and M be two perfect matchings of G such that M M forms a unique cycle of G. Then 0
0
both M and M lie on a directed cycle of Z~ (G). 0
t
3
Theorem 6 Let G be a plane bipartite graph with perfect matchings. Then G is elementary if and only if Z~ (G) is strongly connected. t
Conjecture 7 Let G be a plane elementary bipartite graph. Then the length of each directed cycle of Z~ (G) is equal to the number of faces of G. t
References [1] E. Clar, The Aromatic Sextet (Wiley, London, 1972). [2] I. Gutman and S. J. Cyvin, Advances in the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1990). [3] N. Trinajastic, Chemical Graph Theory (CRC Press, Boca Raton, Florida, 1983). [4] F. Zhang, X. Guo and R. Chen, Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72(1988)405{415. [5] F. Zhang, X. Guo and R. Chen, The connectivity of Z-transformation graphs of perfect matchings of hexagonal systems, Acta Math. Appl. Sinica 4(2)(1988)131{135. [6] F. Zhang and X. Li, Hexagonal systems with forcing edges, Discrete Math. 140(1995)253{263. [7] H. Zhang, The connectivity of Z-transformation graphs of perfect matchings of polyominoes, Discrete Math. 158(1996)257{272. [8] H. Zhang and F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math., in press. [9] H. Zhang and F. Zhang, The rotation graphs of perfect matchings of plane bipartite graphs, Discrete Appl. Math. 73(1997)5{12. [10] H. Zhang and F. Zhang, Block graphs of Z-transformation graphs of perfect matchings of plane elementary bipartite graphs, Ars Combinatoria 53 (1999)309{314. [11] M. Zheng, Perfect matchings in hexagonal systems, Doctoral thesis, Rutgers University, 1992.
4
a;1
Fuji Zhang
b
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China, e-mail address: [email protected] b Department of Mathematics, Xiamen University, Xiamen, Fujian 361005, P. R. China, e-mail address: [email protected]
a
Abstract The total Z-transformation graph and digraph of perfect matchings of a plane bipartite graph are de ned. For a plane elementary bipartite graph it is shown that its total Z-transformation graph and digraph are 2-connected and strongly connected respectively. As an immediate consequence, we have that a plane bipartite graph is elementary if and only if its total Z-transformation digraph is strongly connected. Key words: Plane bipartite graph; Perfect matching; Z-transformation graph
1 Introduction
Let G be a graph with vertex-set V (G) and edge-set E (G). A subset E of E (G) is called a matching of G if no two edges of E share an end-vertex. Further a matching of G is said to be perfect if it covers all vertices of G. An edge of G is said to be allowed if it is contained in a perfect matching of G and forbidden otherwise. Then G is said to be elementary if all allowed edges of G form a connected spanning subgraph of G. It is known that a bipartite graph G is elementary if and only if G is connected and each edge of G is allowed; An elementary bipartite graph with larger than 2 vertices is 2-connected. 0
0
Plane bipartite graphs form an interesting class that contains a family of typical graphs, hexagonal systems, often arising in some real problems. Hexagonal systems with perfect matchings are of chemical signi cance and their topological properties have been extensively studied [1{3,11]. In connection with the 1
supported in part by the national natural science foundation of China (19701014)
Preprint submitted to Elsevier Preprint
7 May 2000
concept of the aromatic sextet for benzenoid hydrocarbons [1], Zhang et al. [4,5] introduced the concept of Z-transformation graphs of perfect matchings of hexagonal systems. By virtue of this concept a complete characterization [6] for hexagonal systems with forcing edges (i.e., those edges belonging to exactly one perfect matching) was given. The present authors in [8] treated general plane bipartite graphs in a uni ed way and also extended naturally the Z-transformation graph of hexagonal systems to general plane bipartite graphs, further in [10] introduced Z-transformation digraph. In mathematical point of view, a more natural de nition for the total Ztransformation (di)graph of a plane bipartite graph involving all the faces is given. In this paper we show that the total Z-transformation graph of a plane elementary bipartite graph is 2-connected; a plane bipartite graph is elementary if and only if the total Z-transformation digraph is strongly connected.
2 Z-transformation graph Let G be a plane bipartite graph with a perfect matching M . A cycle C of G is called an M -alternating cycle if the edges of C appear alternately in M and E (G)nM . An M -alternating path can be de ned similarly. For convenience all the vertices of G are always colored properly black and white; that is, adjacent vertices receive di erent colors. The boundary of an interior face ( nite face) of G is called a ring if it is a cycle. The symmetric di erence of two nite sets A and B is de ned as A B =: (A [ B )n(A \ B ): In the following we restrict our consideration to plane bipartite graphs G with a perfect matching M .
De nition 1 [9]. An M -alternating cycle of G is called a proper M -alternating cycle C if each edge of C belonging to M goes from the white end-vertex to the black end-vertex by the clockwise orientation of C ; otherwise C is known as an improper M -alternating cycle.
De nition 2 Let G be a plane bipartite graph with perfect matchings. The
Z-transformation graph of G, denoted by Z (G), is de ned as a simple graph in which the vertices are the perfect matchings of G and vertices M1 and M2 are adjacent provided the symmetric di erence M1 M2 of the corresponding perfect matchings consists exactly of a ring of G; the Z-transformation digraph, denoted by Z~ (G), is the orientation of Z (G): an edge M1M2 of Z (G) is oriented from M1 to M2 if and only if M1 M2 is a proper M1-alternating ring of G.
For hexagonal systems H the connectivity of Z (H ) is equal to the minimum degree [5]; for polyomino graphs P (or square systems), Z (P ) possesses the same result with the two exceptions [7]. For general plane elementary bipartite 2
graphs G, Z(G) are connected bipartite graphs that may contain cut-vertex [8]. Further, by discussing Z-transformation digraph we obtained the following result.
Theorem 1 [10]. Let G be a plane elementary bipartite graph. Then the blockgraph of the Z-transformation graph of G is a path.
Theorem 2 [10]. Let G be a plane elementary bipartite graph. Then Z~ (G)
contains no directed cycles and has a unique sink t and a unique source s, which belong separately to the two extremal blocks. For any other vertex w, Z~ (G) has a directed path from w to t and a directed path from s to w.
3 Total Z-transformation graph Let G be a plane bipartite graph and C a cycle that is the boundary of a face f of G. The clockwise orientation of C with respect to f is de ned as follows: when going along the direction of cycle C the face f lies on the right side.
De nition 3 Let a plane bipartite graph G have a perfect matching M and
an M -alternating cycle C be the boundary of a face f of G. Then C is called a proper M -alternating boundary if each edge of C belonging to M goes from the white end-vertex to the black end-vertex along the clockwise orientation of C with respect to f ; otherwise C is known as an improper M -alternating boundary.
De nition 4 Let G be a plane bipartite graph with perfect matchings. The
total Z-transformation graph of G, denoted by Z (G), is de ned as a graph in which the vertices are the perfect matchings of G and two perfect matchings M1 and M2 are joined by an edge provided M1 M2 forms a cycle that is the boundary of a face of G; The total Z-transformation digraph of G, denoted Z~ (G), is an orientation of Z (G) such that each edge M1M2 is oriented from M1 to M2 if and only if M1 M2 is a proper M1-alternating boundary of G. t
t
t
Theorem 3 Let G be a plane elementary bipartite graph with more than one cycle. Then the total Z-transformation graph Z (G) is 2-connected. t
Theorem 4 Let a 2-connected plane bipartite graph G have perfect matchings
and do have more than one cycle. Then either Z (G) is 2-connected or each component of Z (G) is 2-connected. t
t
Lemma 5 Let G be a plane elementary bipartite graph. Let M and M be two perfect matchings of G such that M M forms a unique cycle of G. Then 0
0
both M and M lie on a directed cycle of Z~ (G). 0
t
3
Theorem 6 Let G be a plane bipartite graph with perfect matchings. Then G is elementary if and only if Z~ (G) is strongly connected. t
Conjecture 7 Let G be a plane elementary bipartite graph. Then the length of each directed cycle of Z~ (G) is equal to the number of faces of G. t
References [1] E. Clar, The Aromatic Sextet (Wiley, London, 1972). [2] I. Gutman and S. J. Cyvin, Advances in the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1990). [3] N. Trinajastic, Chemical Graph Theory (CRC Press, Boca Raton, Florida, 1983). [4] F. Zhang, X. Guo and R. Chen, Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72(1988)405{415. [5] F. Zhang, X. Guo and R. Chen, The connectivity of Z-transformation graphs of perfect matchings of hexagonal systems, Acta Math. Appl. Sinica 4(2)(1988)131{135. [6] F. Zhang and X. Li, Hexagonal systems with forcing edges, Discrete Math. 140(1995)253{263. [7] H. Zhang, The connectivity of Z-transformation graphs of perfect matchings of polyominoes, Discrete Math. 158(1996)257{272. [8] H. Zhang and F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math., in press. [9] H. Zhang and F. Zhang, The rotation graphs of perfect matchings of plane bipartite graphs, Discrete Appl. Math. 73(1997)5{12. [10] H. Zhang and F. Zhang, Block graphs of Z-transformation graphs of perfect matchings of plane elementary bipartite graphs, Ars Combinatoria 53 (1999)309{314. [11] M. Zheng, Perfect matchings in hexagonal systems, Doctoral thesis, Rutgers University, 1992.
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