A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

S. Bonvicini, T. Pisanski Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Italy, simona.bonvicini@unimore.itUniversity of Primorska, Koper and University of Ljubljana, Ljubljana, Slovenia, Tomaz.Pisanski@fmf.uni-lj.si
Abstract

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by -factor contraction. This correspondence is most useful in the case when it induces a blue and red -factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian -graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian -graphs follows from the fact that one can choose a -factor in any -graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree , that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected -graphs as a subfamily.

Keywords: generalized Petersen graphs, -graphs, Hamiltonian cycles, Eulerian tours, Cayley multigraphs. MSC 2000: 05C45, 05C25, 05C15, 05C76, 05C70, 55R10, 05C60.

1 Introduction.

A graph is Hamiltonian if it contains a spanning cycle (Hamiltonian cycle). To find a Hamiltonian cycle in a graph is an NP–complete problem (see [15]). This fact implies that a characterization result for Hamiltonian graphs is hard to find. For this reason, most graph theorists have restricted their attention to particular classes of graphs.

In this paper we consider cubic graphs. In Section 2 we give a necessary and sufficient condition for a cubic graph to be Hamiltonian. Using this condition we can completely characterize the Hamiltonian -graphs.

The family of -graphs is a generalization of the family of generalized Petersen graphs. In [8], the generalized Petersen graphs were further generalized to -graphs. Let , , be positive integers, with , and . An -graph has vertex-set and edge-set (subscripts are read modulo ). The graph is isomorphic to the graphs , and . It is connected if and only if (see [5]).

For the -graph is known as a generalized Petersen graph and is denoted by . The Petersen graph is . It has been proved that is isomorphic to a generalized Petersen graph if and only if or (see [5]). A connected -graph which is not a generalized Petersen graph is called a proper -graph. Recently, the class of -graphs has been generalized to the class of GI-graphs (see [9]).

It is well known that the Petersen graph is not Hamiltonian. A characterization of Hamiltonian generalized Petersen graphs was obtained by Alspach [2].

Theorem 1 (Alspach, [2]).

A generalized Petersen graph is Hamiltonian if and only if it is not isomorphic to when .

In this paper we develop a powerful theory that helps us extend this result to all -graphs.

Theorem 2.

A connected -graph is Hamiltonian if and only if it is not isomorphic to when .

For the proof the above main theorem, we developed techniques that are of interest by themselves and are presented in the following sections. In particular, we introduce good Eulerian graphs that are similar to lattice diagrams that were originally used by Alspach in his proof of Theorem 1.

Our theory also involves Cayley multigraphs. In Section 4 we show that Cayley multigraphs of degree , that are associated to abelian groups, can be represented as graph bundles [22]. By the results concerning the isomorphisms between Cayley multigraphs (see [10]), we can establish when two graph bundles are isomorphic or not (see Section 4.2). Combining the definition of graph bundles with Theorem 9, we can find a family of connected cubic (multi)graphs that contains the family of connected -graphs as a subfamily (see Section 5).

2 Cubic graph with a -factor and the associated quartic graph with transitions.

A cubic Hamiltonian graph has a -factor. In fact, it has at least three (edge-disjoint) -factors. Namely any Hamiltonian cycle is even and thus gives rise to two -factors and the remaining chords constitute the third -factor. The converse is not true. There are cubic graphs, like the Petersen graph, that have a -factor but are not Hamiltonian. Nevertheless, we may restrict our search for Hamiltonian graphs among the cubic graphs to the ones that possess a -factor. In this section, we give a necessary an sufficient condition for the existence of a Hamiltonian cycle in a cubic graph possessing a -factor .

Let be a connected cubic graph and let be one of its -factors. Denote by the graph obtained from by contracting the edges of . The graph is connected quartic, i.e. regular of degree and might have multiple edges. We say that the quartic graph is associated with and . Since is even and connected, it is Eulerian. A path on three vertices with middle vertex that is a subgraph of is called a transition at . Since any pair of edges incident at defines a transition, there are transitions at each vertex of . For general graphs each vertex of valence gives rise to transitions. In an Euler tour some transitions may be used, others are not used. We are interested in some particular Eulerian spanning subgraphs . Note that any such graph is sub-quartic and the valence at any vertex of is either or . A vertex of valence has therefore transitions, while each vertex of valence has transition. Let be the complementary -factor of in . Note that the edges of are in one-to-one correspondence with the edges of , while the edges of are in one-to-one correspondence with the vertices of . If is an edge of , we denote by the corresponding edge in . If is an edge of the corresponding vertex of will be denoted by . Let and be the end-vertices of and let and be the other edges incident with and similarly and the edges incident with . After contraction of , the vertex is incident with four edges: , , , . By considering the pre images of the six transitions at , they fall into two disjoint classes. Transitions and are non-traversing while the other four transitions are traversing transitions. In the latter case the edge has to be used to traverse from one edge of the pre image transition to the other.

Let be a spanning Eulerian sub-quartic subgraph of . Transitions of carry over . -valent vertices of keep the same six transitions, while each -valent vertex inherits a single transition. We say that is admissible if the transition at each -valent vertex of is traversing.

Let be an admissible subgraph of . A tour in that allows only non-traversing transitions at each -valent vertex of is said to be a tour with allowed transitions. Note that a tour with allowed transitions uses traversing transitions at each -valent vertex of . (We recall that a tour in a graph is a closed walk that traverses each edge of at least once [6]). A tour with allowed transitions might have more than one component.

Lemma 3.

Let be a connected cubic graph with -factor . There is a one-to-one correspondence between -factors of and admissible Eulerian subgraphs of in such a way that the number of cycles of is the same as the number of components of a tour with allowed transitions in .

Proof.

Let be a -factor of and let be an edge of the -factor . Let be the projection of to . We will use the notation introduced above. Hence the edge and its end-vertices and project to the same vertex of . There are two cases:

Case : belongs to . In this case exactly one other edge, say , incident with and another edge, say , incident with belong to . The other two edges ( and ) do not belong to . This means that is a -valent vertex with traversing transition.

Case : does not belong to . In this case both edges and incident with belong to and both edges and incident with belong to . In this case is a -valent vertex with non-traversing transitions.

Clearly, is an admissible Eulerian subgraph. Each component of the tour determined by with transitions gives back a cycle of . The correspondence between and is therefore established.∎

An Eulerian tour in with allowed transitions is said to be good. An admissible subgraph of possessing a good Eulerian tour is said to be a good Eulerian subgraph. In a good Eulerian subgraph there are two extreme cases:

  1. each vertex of is -valent: this means that ; in this case the complementary -factor is a Hamiltonian cycle and no edge of is used;

  2. each vertex of is -valent: this means that is a good Hamiltonian cycle in . In this case together with the pre images of edges of in form a Hamiltonian cycle.

Theorem 4.

Let be a connected cubic graph with -factor . Then is Hamiltonian if and only if contains a good Eulerian subgraph .

Proof.

Clearly is Hamiltonian if and only if it contains a -factor with a single cycle. By Lemma 3, this is true if and only if is an admissible Eulerian subgraph possessing an Eulerian tour with allowed transitions. But this means is good.∎

Corollary 5.

Let be a connected cubic graph with -factor . Finding a good Eulerian subgraph of is NP-complete.

Proof.

Since finding a good Eulerian subgraph is equivalent to finding a Hamiltonian cycle in a cubic graph, and the latter is NP-complete [15], the result follows readily.∎

Also in [14] Eulerian graphs are used to find a Hamiltonian cycle (and other graph properties), but our method is different.

The results of this section may be applied to connected -graphs. The obvious 1-factor of an -graph consists of spokes. Let denote the quotient . We will call the quartic graph associated with .

Corollary 6.

Let be a connected -graph and let be its associated quartic graph. Then is Hamiltonian if and only if contains a good Eulerian subgraph .

3 Special -factors and their applications.

Let be a cubic graph, a -factor and the complementary -factor of in . Define an auxiliary graph having cycles of as vertices and having two vertices adjacent if and only if the corresponding cycles of are joined by one ore more edges of . If an edge of is a chord in one of the cycles of , then the graph has a loop. We shall say that the -factor is special if the graph is bipartite. A cubic graph with a special -factor will be called special. If is a special -factor of , then the edges of join vertices belonging to distinct cycles of , since is loopless.

Theorem 7.

Let be a connected cubic graph and let be one of its -factors and its associated quartic graph. Then admits a -factorization into a blue and red -factor in such a way that the traversing transitions are exactly color-switching and non-traversing transitions are color-preserving if and only if and are special.

Proof.

Assume that is a special -factor of . Since is bipartite, we can bicolor the vertices of the bipartition: let one set of the bipartition be blue and the other red. This coloring induces a coloring on the edges of : for every blue vertex (respectively, red vertex) of we color in blue (respectively, in red) the edges of the corresponding cycle of . Since the edges of are in one-to-one correspondence with the edges of , we obtain a -factorization of into a blue and red -factor. Since is special, the edges of are incident with vertices of belonging to cycles of with different colors (a blue cycle and a red cycle). Therefore, a traversing transitions is color-switching and a non-traversing transition is color-preserving.

Conversely, assume that has a blue and red -factorization such that the traversing transitions are color-switching and non-traversing transitions are color-preserving. Since the edges of are in one-to-one correspondence with the edges of , we can partition the cycles of into red cycles and blue cycles. Since the traversing transitions are color-switching and non-traversing transitions are color-preserving, the edges of are incident with edges belonging to cycles of different colors. This means that the graph is bipartite, hence is special.∎

Proposition 8.

Let and be special and let be any Eulerian subgraph of the associated quartic graph with a blue and red -factorization. Then is admissible if and only if each -valent vertex is incident with edges of different colors.

Proof.

An Eulerian subgraph is admissible if and only if each -valent vertex in is incident with edges forming a traversing transition at . By Theorem 7, a traversing transition is color-switching. Hence, is admissible if and only if the edges incident with have different colors.∎

Note that quartic graphs with a given -factorization can be put into one-to-one correspondence with special cubic graphs.

Theorem 9.

Any special cubic graph with a special -factor gives rise to the associated quartic graph with a blue and red -factorization. However, any quartic graph with a given -factorization determines back a unique special cubic graph by color-preserving splitting vertices and inserting a special -factor.

Proof.

By Theorem 7, a special cubic graph with a special -factor gives rise to the graph admitting a blue and red -factorization.

Conversely, it is well known that every quartic graph possesses a -factorization, i.e. the edges of can be partitioned into a blue and red -factor. We use the blue and red -factors of to construct a cubic graph as follows: put in a copy of the blue -factor and a copy of the red -factor; construct a -factor of by joining vertices belonging to distinct copies. It is straightforward to see that and are special.∎

We will now apply this theory to the -graphs. In Section 7 we will see that this theory allow us to find a Hamiltonian cycle in a proper -graph and also to find a family of special cubic graphs that contains the family of -graphs as a subfamily (see Section 5).

Let be an -graph. A vertex (respectively, ) is called an outer vertex (respectively, an inner vertex). An edge of type (respectively, of type ) is called an outer edge (respectively, an inner edge). An edge is called a spoke. The spokes of determine a -factor of . The set of outer edges is said the outer rim, the set of inner edges is said the inner rim. As a consequence of the results proved in [5], the following holds.

Proposition 10.

Let , , , , be an -graph. Set and . Then and . Moreover is connected if an only if and is coprime with . It is proper if and only if and are different from .

Proof.

The integer satisfies the inequality , since is a divisor of and , ; whence . By the results proved in [5], is connected if and only if . Since and , the relation can be written as , whence and is coprime with . Also the converse is true, therefore is connected if and only if and is coprime with . A connected -graph is a generalized Petersen graph if and only if or (see [5]). By the previous results, is a generalized Petersen graph if and only if or . The assertion follows.∎

The smallest proper -graphs are and . It is straightforward to see that the following result holds.

Lemma 11.

Let be the -factor determined by the spokes of and its associated quartic graph. Then is special, the graph is a circulant multigraph , the blue edges of correspond to the inner rim and the red edges to the outer rim of .

In the next section we introduce a class of graphs and later show that it contains as its subclass.

4 Graphs .

Let be a group in additive notation with identity element . Let be a list of not necessarily distinct elements of satisfying the symmetry property . The Cayley multigraph associated with and , denoted by , is an undirected multigraph having the elements of as vertices and edges of the form with , . If is a cyclic group of order , then is a circulant multigraph of order . A Cayley multigraph is regular of degree (in determining , each element of is considered according to its multiplicity in ). It is connected if an only if is a set of generators of . If the elements of are pairwise distinct, then is a simple graph and we will speak of Cayley graph. We are interested in connected Cayley multigraphs of degree . In this case we write as the list . A circulant multigraph of order will be denoted by . If , with , is an involution of or the trivial element, then means that the element appears twice in the list . Consequently, the associated Cayley multigraph has multiple edges or loops. We will denote by the order of . We will show that the Cayley multigraphs defined on a suitable abelian group (and in particular the circulant multigraphs ) can be given a different interpretation in terms of graphs (see Figure 1) defined as follows.

Definition 1.

Let , be integers. Let be the graph with vertex-set and edge-set (the superscripts are read modulo , the subscripts are read modulo ).

The graph has edges of type , or . An edge of type will be called horizontal. An edge of type will be called vertical, an edge of will be called diagonal. For , we say that the edges are horizontal and diagonal (a diagonal edge is an edge of type ). For or , each diagonal edge is a loop. For or and =, , the graph has multiple edges. For the other values of , , , the graph is a simple graph. A simple graph is a graph bundle with a cycle fiber over a cycle base ; the parameter represents an automorphism of the cycle that shifts the cycle steps (see [22] for more details on graph bundles). In literature a simple graph is also called -pseudo-cartesian product of two cycles (see for instance [13]). The definition of suggests that the graph is isomorphic to . The existence of this isomorphism can be also obtained from the following statement.

Figure 1: The graph is embedded into torus with quadrilateral faces; it has a blue and red -factorization: the vertical and diagonal edges form the blue -factor, the horizontal edges form the red -factor.
Proposition 12.

Let be a connected Cayley multigraph of degree , where is an abelian group, and . Then for some integer , , if and only if . Consequently, can be represented as the graph or .

Proof.

We show that and are isomorphic. Since and are generators of , the elements of can be written in the form , where , . Hence we can describe the elements of by the left cosets of the subgroup in . By this representation, we can see that the endvertices of an edge of belong to the same left coset of in ; the endvertices of an edge belong to distinct left cosets of in . Therefore, if and only if , since is connected and . Hence we can set and . The map defined by is a bijection between and . Moreover, if , are adjacent vertices of , i.e., and (or ), then , (or ) are adjacent vertices of . In particular, if and , then , are adjacent vertices of . It is thus proved that is an isomorphism between and . If we replace the element by its inverse , then is the graph .∎

In what follows, we show that for there exists a Cayley multigraph on a suitable abelian group that satisfies Proposition 12, i.e., for every the graph can be represented as a Cayley multigraph. The proof is particularly easy when ; ; or . For these cases, the following holds.

Proposition 13.

The graph , with , , is the circulant multigraph ; . The graph , with , is the Cayley multigraph , , . The graph , with , is the circulant multigraph .

Proof.

For the graph we apply Proposition 12 with , , . For the graph we apply Proposition 12 with , , . For the graph we apply Proposition 12 with , , . ∎

The following lemmas concern the graph with , and . They will be used in the proof of Proposition 16.

Lemma 14.

Let be an integer and let be a divisor of . Let be the residue classes modulo . Every equivalence class whose representative is coprime with contains at least one integer , , such that .

Proof.

The assertion is true if (we set ). We consider . Let be an equivalence class modulo with and . If is coprime with , then we set and the assertion follows. We consider the case . We denote by the set of distinct prime numbers dividing . We denote by (respectively, by ) the subset of containing the prime numbers dividing (respectively, ). Since is a divisor of (respectively, ), the set (respectively, ) is non-empty. Since and are coprime, the subsets , are disjoint. We set . The set might be empty. We denote by the product of the prime numbers in (if is empty, then we set ) and consider the integer . We show that . Note that . More specifically, , whence . Hence , since and . One can easily verify that , since no prime number in can divide . The assertion follows.∎

Lemma 15.

Let , and be the cyclic group of integers modulo , where is a divisor of . Let be the cyclic subgroup of generated by the integer and let , be left cosets of in . If , are congruent modulo , then and represent the same subset of .

Proof.

Set with and with and . Since , then also . Hence the integers generate the same cyclic subgroup of of order . Since , each set , consists of exactly distinct elements of , namely, , . Therefore, to prove that , it suffices to show that every element of is contained in . Consider . Since , we can write , whence . Since , we can set , with . Hence , that is, . The assertion follows.∎

Proposition 16.

Let , , , with . The cyclic group of integers modulo contains an integer such that and . The graph can be represented as the Cayley graph . In particular, if then can be represented as the circulant graph .

Proof.

Set . Since , the integer is coprime with . Hence, belongs to an equivalence class whose representative is coprime with . By Lemma 14, the class contains an integer , , such that . Consider the cyclic group . Since , , the integer and belong to . Hence we can apply Lemma 15 with , and find that , i.e., there exists an integer such that . The integer is coprime with , since . The assertion follows from Proposition 12, by setting , , . Note: if , then is the cyclic group of order , , and is the circulant graph .∎

The result that follows is based on a well-known consequence of the Chinese Remainder Theorem. More specifically, it is known that if , , and are positive integers, with , then the equation admits a solution if and only if is a divisor of and in this case is a solution to the equation. The following holds.

Proposition 17.

Let , and . If , then there exists an integer , , such that and . The graph , with , is isomorphic to the graph , , , where . The graph is isomorphic to the graph .

Proof.

We prove the assertion for , and . The existence of the integer follows from Proposition 16. By the same proposition, we can represent the graph as . We apply Proposition 12 by setting , and . Note that , as is coprime with and . Whence the element has order and . By Proposition 12, for some integer , . The integer is a solution to the equation . By the Chinese Remainder Theorem, . An easy calculation shows that . It is straightforward to see that and , are isomorphic. Hence the assertion follows. For the remaining values of , , , we represent the graph as the Cayley multigraph in Proposition 13 and use Proposition 12. Note: if , then ; if , then set , and apply Proposition 12.∎

4.1 Fundamental -factorization of .

From the definition of one can see that the horizontal edges form a -factor (the red -factor) whose complementary -factor in is given by the vertical and diagonal edges (the blue -factor). We say that the red and blue -factor constitute the fundamental -factorization of . A graph can be represented as a Cayley multigraph , where and can be defined as in Proposition 13 or 16. From the proof of the propositions, one can see that the set of horizontal edges of is the set , the set of vertical and diagonal edges is the set . The edges in will be called the -edges, the edges in the set will be called the -edges. The following result holds.

Proposition 18.

The red -factor of has exactly cycles of length consisting of -edges. The blue -factor of has exactly cycles of length consisting of -edges.

Proof.

It is straightforward to see that the red -factor has horizontal cycles of length (if , then each cycle is a loop; if , then each cycle is a dipole with parallel edges). By the previous remarks, each cycle consists of -edges. The blue -factor of corresponds to the red -factor of the graph in Proposition 17. Hence it has cycles of length consisting of -edges.∎

4.2 Isomorphisms between graphs.

We wonder whether two graphs and are isomorphic. Our question is connected to the following well-known problem [10]: given two isomorphic Cayley multigraphs , , determine whether the groups , are necessarily A-isomorphic, that is, there exists an isomorphism between and that sends the set onto the set ( is called an A-isomorphism and is denoted by ). Adám [1] considered this problem for circulant graphs and formulated a well-known conjecture which was disproved in [12]. He conjectured that two circulant graphs , are isomorphic if and only if there exists an integer , , such that . Even though the conjecture was disproved, there are some circulant graphs that verify it (see for instance [19]). In [10] the problem is studied for Cayley multigraphs of degree which are associated to abelian groups. The results in [10] are described in terms of Adám isomorphisms. We recall that an Adám isomorphism between the graphs , is an isomorphism of type , where is an A-isomorphism and is the automorphism of the graph defined by for every vertex . Since the graphs can be represented as Cayley multigraphs, we can extend the notion of Adám isomorphism to the graphs . We will say that the graphs , are Adám isomorphic if the corresponding Cayley multigraphs , , respectively, are Adám isomorphic (, are described in Proposition 13 or 16). The following statements hold.

Proposition 19.

Every Adám isomorphism between the graphs ,