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Characterizations of bipartite and Eulerian partial duals of ribbon graphs
Discrete Mathematics xxx (xxxx) xxx
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Characterizations of bipartite and Eulerian partial duals of ribbon graphs ∗
Qingying Deng a , Xian’an Jin a , , Metrose Metsidik b a b
School of Mathematical Sciences, Xiamen University, PR China College of Mathematical Sciences, Xinjiang Normal University, PR China
article
info
Article history: Received 29 November 2017 Received in revised form 5 August 2019 Accepted 15 August 2019 Available online xxxx Keywords: Ribbon graph Partial dual Medial graph Direction Bipartite Eulerian
a b s t r a c t Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions. © 2019 Elsevier B.V. All rights reserved.
1. Introduction A graph is said to be Eulerian if the degree of each of its vertices is even. In this paper, we do not require that an Eulerian graph must be connected, and such graphs are sometimes called even graphs. A graph is said to be bipartite if it does not contain cycles of odd lengths. A ribbon graph is a new form of the old cellularly embedded graph and the signed rotation system [11], which makes the concept of partial duality possible. We do not require graphs to be connected throughout the whole paper unless otherwise stated. The geometric dual, G∗ , of a cellularly embedded graph G is a fundamental concept in graph theory and also appeared in other branches of mathematics. To unify several Thistlethwaite’s theorems [3–5] on relations between the Jones polynomial of virtual links and the topological Tutte polynomial of ribbon graphs constructed from virtual links in different ways, S. Chmutov, in [2], introduced the concept of partial dual of a cellularly embedded graph in terms of ribbon graphs. Roughly speaking, a partial dual is obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. If G = (V , E) is a ribbon graph and A ⊂ E, we denote by GA the partial dual of G with respect to A. Note that GE = G∗ , G and G∗ as surfaces have the same orientability and genuses. Although GA has the same orientablity as G, the genus may change. For example, a partial dual of a plane graph may not be plane again. The following theorem is well-known, see, for example, Theorem 34.4 of [8] or example 10.2.10 of [1]. Theorem 1.1. Let G be a plane graph. Then G is Eulerian if and only if G∗ is bipartite. ∗ Corresponding author. E-mail addresses: [email protected] (Q. Deng), [email protected] (X. Jin), [email protected] (M. Metsidik). https://doi.org/10.1016/j.disc.2019.111637 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 1. Relations among above four families of embedded graphs.
In [7], Huggett and Moffatt extended the above connection between Eulerian and bipartite plane graphs from geometrical duality to partial duality. A ribbon graph is said to be even-face if each of its boundary components is of even degree. Note that a bipartite ribbon graph is always an even-face one but the converse is not always true but holds for plane graphs. For example, the graph consisting of longitude and latitude circles in a torus (one vertex and two loops) is even-face, but not bipartite. We shall use even-face to replace bipartite and then we have: for a ribbon graph G, G is Eulerian if and only if G∗ is an even-face graph. Extending this to partial duality, it is easily observed that: Observation 1.2. Eulerian.
c
Let G be a ribbon graph, A ⊂ E(G) and Ac = E(G) − A. Then GA is an even-face graph if and only if GA is
A checkerboard coloring of a cellularly embedded graph is an assignment of the color black or color white to each face such that adjacent faces receive different colors. A cellularly embedded graph is said to be checkerboard colorable if it has a checkerboard coloring. The following lemma is well-known. Lemma 1.3.
A cellularly embedded graph G is bipartite if and only if its geometric dual G∗ is checkerboard colorable.
The relations among above four families of embedded graphs is shown in Fig. 1. In [7], Huggett and Moffatt further determined which subsets of edges in a plane graph give rise to bipartite partial duals. In this paper, we extend their result from plane graphs to ribbon graphs and obtain the first main result. Theorem 1.4. Let G be an ribbon graph and A ⊂ E(G). Then GA is bipartite if and only if A is the set of c-edges arising from ˆ an all-crossing direction of G m , the modified medial graph (which is defined in Section 2.2) of G. In [7], Huggett and Moffatt also found a sufficient condition for a subset of edges to give rise to an Eulerian partial dual in terms of all-crossing directions of medial graphs. In [10], Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. In this paper, we find more simple directions, i.e. crossing-total directions, of medial graph to characterize all Eulerian partial duals of any ribbon graph and obtain our second main result. Theorem 1.5. Let G be a ribbon graph and A ⊂ E(G). Then GA is Eulerian if and only if A is union of the set of all d-edges and the set of some t-edges arising from a crossing-total direction of Gm . 2. Preliminaries In this section we only give necessary preliminaries and refer the readers to the monograph [6] for details on ribbon graphs, their equivalent notions such as arrow presentations, and their partial duals. 2.1. Three types of line segments Definition 2.1. A ribbon graph G = (V (G), E(G)) is a (possibly nonorientable) surface with boundary represented as the union of two sets of topological discs: a set V (G) of vertices and a set E(G) of edges such that:
• the vertices and edges intersect in disjoint line segments; • each such line segment lies on the boundary of precisely one vertex and precisely one edge; • every edge contains exactly two such line segments. Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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We shall call these line segments in Definition 2.1 common line segments as in [9] since they belong to boundaries of both vertex discs and edge discs. Let G be (an abstract or a ribbon) graph and v ∈ V (G). We denote by dG (v ) the degree of v in G, i.e. the number of half-edges incident with v . Definition 2.2 ([9]). Let G be a ribbon graph, v ∈ V (G) and e ∈ E(G). By deleting the common line segments from the boundary of v , we obtain dG (v ) disjoint line segments, we shall call them vertex line segments. By deleting common line segments from the boundary of e, we obtain two disjoint line segments, we shall call them edge line segments. On a boundary component of a ribbon graph, vertex line segments and edge line segments alternate. The number of vertex line segments (equivalently, edge line segments) on a boundary component of a ribbon graph is called the degree of the boundary component. As an example, in Fig. 2(a), all vij ’s are vertex line segments, all eij ’s are edge line segments and all e′ij ’s are common line segments. v11 e32 v23 e21 v31 e12 v22 e31 v12 e52 is a boundary component of degree 5. The following lemma is no more than an observation. Lemma 2.3.
Let G be a ribbon graph and A ⊂ E(G). Let GA be the partial dual of G with respect to A. Then
(1) there is a natural bijection between vertex line segments of G and GA ; common line segments and edge line segments of edges inside A exchange after taking the dual; and common line segments and edge line segments of edges outside A keep unchanged after taking the dual. (2) the boundary components of G consist of vertex line segments and edge line segments; the boundary component of GA consist of vertex line segments, common line segments of edges inside A and edge line segments of edges outside A. (3) the boundary components of vertex discs of G consist of vertex line segments and common line segments; the boundary components of vertex discs of GA (i.e. circles of the arrow presentation of GA ) consist of vertex line segments, common line segments of edges outside A and edge line segments of edges inside A. (4) there is a natural bijection between boundary components of G and boundary components of (V (GA ), Ac ). Proof. We use Fig. 2 to show the lemma. For (1), see (a) and (d). (b) and (c) are intermediate steps from (a) to (d). For (2), see (a) and (e). For (3), see (a) and (d) again. For (4), see (a) and (f). □ As shown in Fig. 2(e) and (h), there is a bijection between boundary components of GA and vertex boundaries of GA and Observation 1.2 is thus established.
c
2.2. Medial graphs and directions Let G be a cellularly embedded graph in Σ . We construct its medial graph Gm by placing a vertex v (e) on each edge e of G and then for each face f with boundary v1 e1 v2 e2 · · · vd(f ) ed(f ) , drawing the edges (v (e1 ), v (e2 )), (v (e2 ), v (e3 )), · · ·, (v (ed(f ) ), v (e1 )) non-intersected in a natural way inside the face f . Note that Gm is also a cellularly embedded graph in Σ and a 4-regular one. We can color the faces of the medial graph Gm containing a vertex of G black and the remaining faces white, obtaining the canonical checkerboard colouring of Gm . See Fig. 3(a) and (b) for an example. A crossing-total direction of Gm is an assignment of an orientation to each edge of Gm in such a way that for each vertex v (e) of Gm , edges incident with v (e) are ‘‘in,in,out,out", ‘‘in,in,in,in" or ‘‘out,out,out,out" oriented in cyclic order with respect to v (e). See Fig. 3(c) for an example. If Gm is equipped with the canonical checkerboard colouring and a fixed crossing-total direction, then we can partition the vertices of Gm into three classes, called c-vertices, d-vertices and t-vertices, respectively according to the scheme shown in Fig. 4. Note that in Fig. 4, we draw the medial graph Gm of a ribbon graph G as a cellularly embedded graph, not as a ribbon graph. There is a natural bijection e ↔ v (e) between edges of G and vertices of Gm . The corresponding edges of G are called c-edges, d-edges and t-edges, respectively. See Fig. 3(d) for an example. Note that Gm = (G∗ )m and the canonical checkerboard colouring of (G∗ )m can be obtained from that of Gm by switching colours black and white. Lemma 2.4.
Let G be a cellularly embedded graph and e ∈ E(G). Under a fixed crossing-total direction of Gm , we have:
(1) e is a c-edge in G if and only if e is a d-edge in G∗ ; (2) e is a t-edge in G if and only if e is a t-edge in G∗ . In particular, a bridge e of a cellularly embedded graph G is always a d-edge in any fixed crossing-total direction of Gm . In [7], Huggett and Moffatt introduced the notion of all-crossing direction of Gm and then characterized all bipartite partial duals of a plane graph using all-crossing directions. An all-crossing direction is a crossing-total direction without t-vertices. Note that we can draw an orientable ribbon graph in the plane without twist in each vertex disc and each edge disc, and draw a non-orientable ribbon graph in the plane without twist in each vertex disc and with at most one twist in each edge disc. That is, for a given choice of orientation (clockwise direction) for each vertex disc, there are two types of edges Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 2. Illustration of Lemma 2.3: (a) a ribbon graph G and its common line segments, vertex line segments and edge line segments, (b) to (d) the construction of GA with A = {1, 2, 5} and corresponding line segments, (e) the boundary component of GA , (f) the boundary component of (V (GA ), Ac ), c (g) the marked spanning subgraph (V (G), Ac ) and (h) the arrow presentation of GA .
(twisted or untwisted) in a non-orientable ribbon graph. When forming the medial graph Gm , we must orient the vertex disk for each v (e) ∈ V (Gm ). If e is an untwisted edge of G with ends w , x we orient the disc for v (e) in the way that agrees with the orientations for the discs of w and x in G. If e is a twisted edge we choose an orientation for the disc for v (e); it will agree with the orientation of one end of e, say w , and disagree with the orientation of x. We must then apply twists to the two edges of the medial graph incident with v (e) that correspond to x. Some edges of the medial graph may be given two twists that cancel out. Thus, if there are τ twisted edges in G, then there are at most 2τ twisted edges in Gm . We add a 2-valent vertex (denoted by short slash) to each twisted edge in Gm , and denote the new graph by ˆ Gm and call it the modified medial graph. See Fig. 5(b). If Gm is equipped with the canonical checkerboard colouring, then we can obtain the checkerboard colouring of ˆ Gm naturally by neglecting extra artificial 2-valent vertices. If we direct edges of ˆ Gm such that every 4-valent vertex is c-vertex or d-vertex and every 2-valent vertex has ‘‘in, in’’ or ‘‘out, out’’ direction, we call the direction of ˆ Gm an all-crossing direction of ˆ Gm . See Fig. 5(c). When there are no such 2-valent vertices in ˆ Gm , ˆ Gm reduces to Gm . Fig. 5 gives an example of bipartite partial dual of G based on an all-crossing direction of ˆ Gm . Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 3. (a) a plane graph G, (b) its medial graph Gm with canonical checkerboard coloring, (c) a crossing-total direction of Gm , and (d) the corresponding {c , d, t }-edges of G.
Fig. 4. c-vertex, d-vertex and t-vertices.
Fig. 5. (a) a non-orientable ribbon graph G, (b) the modified medial graph ˆ Gm embedded in the ribbon graph G, (c) an all-crossing direction of ˆ Gm , and (d) arrow presentation of the bipartite partial dual G{5} .
2.3. Graph states Let Gm be a canonically checkerboard coloured medial graph and v (e) ∈ V (Gm ). There are two ways to split v (e) into two vertices of degree 2 as shown in Fig. 6. We call them white smoothing and black smoothing respectively. Let G be a cellularly embedded graph in Σ . A state S of G is a choice of white smoothing and black smoothing at each vertex of Gm . After splitting each vertex according to the state S, we obtain a set of disjoint closed cycles in Σ , call them state circles of S. We denote the number of state circles of S by c(S). A state is called even if each of its state circles is a cycle of even length. We denote the set of even states of G by E S . Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 6. White and black smoothings.
Fig. 7. Renaming vertex line segments for GA in Fig. 2.
Let A ⊂ E(G) and VA = {v (e)|e ∈ A} ⊂ V (Gm ). A state SA of G associated with A is the choice of choosing white smoothing for each vertex of VA and black smoothing for each vertex of V (Gm )\VA . Lemma 2.5. Let G ⊂ Σ be a cellularly embedded graph and A ⊂ E(G). Let SA be the graph state of G associated with A. Then there is a bijection between the vertices (of degree 2) of state circles of SA and the marking arrows (ignoring directions) of the arrow presentation of GA . Proof. Compare Fig. 6 with Definition of arrow presentation in [6]. □ Lemma 2.5 is a key observation that will be used in the proofs of the main results (Theorems 1.4 and 1.5). 3. Proofs Proof of Theorem 1.4. (H⇒) Let (X , Y ) be the bipartition of V (GA ). Suppose that X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl }. We rename each vertex line segment of GA as follows: if the vertex line segment vrs is on the boundary of xi , it will be renamed xi ; if it is on the boundary of yj , it will be renamed yj . An example is given in Fig. 7, where X = {x1 , x2 }, Y = {y1 , y2 }, ∂ (x1 ) ⊃ {v14 , v13 }, ∂ (y1 ) ⊃ {v11 , v12 }, ∂ (x2 ) ⊃ {v22 , v23 , v31 }, ∂ (y2 ) ⊃ {v21 , v33 , v32 }. See also Fig. 8(a), (a′ ), (a′′ ) and (a′′′ ). There is a natural bijection between vertex line segments of G and GA by Lemma 2.3 (1). We rename vertex line segments of G accordingly. Note that there is a natural bijection between vertex line segments of G and the edges of Gm . Thus we label each edge of Gm with X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl }. Then we label each edge of ˆ Gm with X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl } as Gm except that there are two identical labels for each twisted edge of Gm . In ˆ Gm , for an untwisted edge eh ∈ A, the label of edges incident to veh is shown as in Fig. 8(b); for an untwisted edge eh ∈ Ac , the label of edges incident to veh is shown as in Fig. 8 (b′ ); for a twisted edge eh ∈ A, the label of edges incident to veh is shown as in Fig. 8 (b′′ ); for a twisted edge eh ∈ Ac , the label of edges incident to veh is shown as in Fig. 8 (b′′′ ). According to the canonical checkerboard colouring of ˆ Gm , we direct the edge labelled xi and yj of ˆ Gm such that the black colour is on their right and left (using the orientation at the appropriate vertex of G to define right and left) along the directions, respectively, as shown in Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). It is obvious that the direction is an all-crossing direction of ˆ Gm . Furthermore each edge of A is a c-edge and each edge of Ac is a d-edge as shown in Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). (⇐H) Conversely, let ˆ Gm be equipped with the canonical checkerboard colouring. Let O be an all-crossing direction ˆ of Gm such that A is the set of c-edges arising from O. Then Ac is the set of d-edges. We label each edge of ˆ Gm with x if the black colour is on the right as we follow the direction of the edge, and with y if the black colour is on the left. See Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). Note that we can label x or y the twisted edge of ˆ Gm consistently. Let SA be the state of G associated with A. State circles of SA correspond to circles of arrow presentation of GA as shown in Fig. 8(a), (a′ ), (a′′ ) and Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 8. Proof of Theorem 1.4.
(a′′′ ). The circles labelled x and circles labelled y correspond to the circles in SA , i.e., the circles of the arrow presentation of GA . Then circles labelled x and circles labelled y form vertex bipartition of GA . Thus GA is bipartite. □ As a direct consequence, we have Corollary 3.1. Let G be a ribbon graph and A ⊂ E(G). Then GA is checkerboard colorable if and only if A is the set of d-edges arising from an all-crossing direction of ˆ Gm , the modified medial graph of G. Corollary 3.2. Let G be an orientable ribbon graph and A ⊂ E(G). Then GA is bipartite if and only if A is the set of c-edges arising from an all-crossing direction of Gm . Remark 3.3. Let G1 (left) and G2 (right) be non-orientable ribbon graphs shown in Fig. 9. They are dual to each other. (G1 )m and (G2 )m both have only one straight-ahead walks, the all-crossing direction of (G1 )m has no c-vertices and the all-crossing direction of (G2 )m has no d-vertices. This shows that Corollary 3.2 does not hold for non-orientable ribbon graphs. Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 9. Examples which show that Corollary 3.2 does not hold for non-orientable ribbon graphs.
We now prove our second main but simpler result, a characterization of Eulerian partial duals of a ribbon graph in terms of crossing-total directions of its medial graph. But we first need the following crucial lemma. Lemma 3.4.
Let G be a ribbon graph and A ⊂ E(G). Then GA is Eulerian if and only if SA of G associated with A is even.
Proof. The degree of a vertex of GA is equal to the number of marking arrows of the corresponding circle of arrow presentation of GA . By Lemma 2.5, it is exactly the length of corresponding state circle of SA of G. Thus GA is Eulerian if and only if SA is even. □ Proof of Theorem 1.5. (H⇒) By Lemma 3.4, SA is even. So we can give an alternating (clockwise and counter-clockwise) orientation to each circle of SA . Transferring them to Gm , we obtain a crossing-total direction of Gm . It is easy to see that each edge of A is either a d-edge or a t-edge and each edge of Ac is either a c-edge or a t-edge. Thus A is the union of the set of all d-edges and some t-edges arising from a crossing-total direction of Gm . (⇐H) Conversely, if A is the union of the set of all d-edges and some t-edges arising from a crossing-total direction of Gm . Then SA is an even state of G since the edge orientations of each state circle of SA are alternating between clockwise and counterclockwise orientation. Thus GA is Eulerian by Lemma 3.4. □ Using Observation 1.2 and Lemma 2.4, we obtain: Corollary 3.5. Let G be a ribbon graph and A ⊂ E(G). Then GA is even-face graph if and only if A is the union of the set of all c-edges and the set of some t-edges arising from a crossing-total direction of Gm . Declaration of competing interest The authors declare that there is no conflict of interest in this paper. Acknowledgments This work is supported by NSFC, PR China (No. 11671336) and the Fundamental Research Funds for the Central Universities, PR China (No. 20720190062). We thank the referees and editor for helpful suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
J.A. Bondy, U.S.R. Murty, Graph Theory, in: Graduate Texts in Mathematics, vol. 244, Springer, Berlin, 2008. S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory Ser. B 99 (2009) 617–638. S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, Moscow Math. J. 7 (3) (2007) 409–418. S. Chmutov, J. Voltz, Thistlethwaite’s theorem for virtual links, J. Knot Theory Ramif. 17 (10) (2008) 1189–1198. O. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin, N. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2) (2008) 384–399. J.A. Ellis-Monaghan, I. Moffatt, Graphs on Surfaces: Dualities, Polynomials, and Knots, in: Springer Briefs in Mathematics, 2013. S. Huggett, I. Moffatt, Bipartite partial duals and circuits in medial graphs, Combinatorica 33 (2) (2013) 231–252. J. van Lint, R. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 2001. M. Metsidik, Characterization of some properties of ribbon graphs and their partial duals (Ph.D. thesis), Xiamen University, 2017. M. Metsidik, X. Jin, Eulerian partial duals of plane graphs, J. Graph. Theory 87 (4) (2018) 509–515. B. Mohar, C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, Baltimore, MD, 2001.
Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Characterizations of bipartite and Eulerian partial duals of ribbon graphs ∗
Qingying Deng a , Xian’an Jin a , , Metrose Metsidik b a b
School of Mathematical Sciences, Xiamen University, PR China College of Mathematical Sciences, Xinjiang Normal University, PR China
article
info
Article history: Received 29 November 2017 Received in revised form 5 August 2019 Accepted 15 August 2019 Available online xxxx Keywords: Ribbon graph Partial dual Medial graph Direction Bipartite Eulerian
a b s t r a c t Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions. © 2019 Elsevier B.V. All rights reserved.
1. Introduction A graph is said to be Eulerian if the degree of each of its vertices is even. In this paper, we do not require that an Eulerian graph must be connected, and such graphs are sometimes called even graphs. A graph is said to be bipartite if it does not contain cycles of odd lengths. A ribbon graph is a new form of the old cellularly embedded graph and the signed rotation system [11], which makes the concept of partial duality possible. We do not require graphs to be connected throughout the whole paper unless otherwise stated. The geometric dual, G∗ , of a cellularly embedded graph G is a fundamental concept in graph theory and also appeared in other branches of mathematics. To unify several Thistlethwaite’s theorems [3–5] on relations between the Jones polynomial of virtual links and the topological Tutte polynomial of ribbon graphs constructed from virtual links in different ways, S. Chmutov, in [2], introduced the concept of partial dual of a cellularly embedded graph in terms of ribbon graphs. Roughly speaking, a partial dual is obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. If G = (V , E) is a ribbon graph and A ⊂ E, we denote by GA the partial dual of G with respect to A. Note that GE = G∗ , G and G∗ as surfaces have the same orientability and genuses. Although GA has the same orientablity as G, the genus may change. For example, a partial dual of a plane graph may not be plane again. The following theorem is well-known, see, for example, Theorem 34.4 of [8] or example 10.2.10 of [1]. Theorem 1.1. Let G be a plane graph. Then G is Eulerian if and only if G∗ is bipartite. ∗ Corresponding author. E-mail addresses: [email protected] (Q. Deng), [email protected] (X. Jin), [email protected] (M. Metsidik). https://doi.org/10.1016/j.disc.2019.111637 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 1. Relations among above four families of embedded graphs.
In [7], Huggett and Moffatt extended the above connection between Eulerian and bipartite plane graphs from geometrical duality to partial duality. A ribbon graph is said to be even-face if each of its boundary components is of even degree. Note that a bipartite ribbon graph is always an even-face one but the converse is not always true but holds for plane graphs. For example, the graph consisting of longitude and latitude circles in a torus (one vertex and two loops) is even-face, but not bipartite. We shall use even-face to replace bipartite and then we have: for a ribbon graph G, G is Eulerian if and only if G∗ is an even-face graph. Extending this to partial duality, it is easily observed that: Observation 1.2. Eulerian.
c
Let G be a ribbon graph, A ⊂ E(G) and Ac = E(G) − A. Then GA is an even-face graph if and only if GA is
A checkerboard coloring of a cellularly embedded graph is an assignment of the color black or color white to each face such that adjacent faces receive different colors. A cellularly embedded graph is said to be checkerboard colorable if it has a checkerboard coloring. The following lemma is well-known. Lemma 1.3.
A cellularly embedded graph G is bipartite if and only if its geometric dual G∗ is checkerboard colorable.
The relations among above four families of embedded graphs is shown in Fig. 1. In [7], Huggett and Moffatt further determined which subsets of edges in a plane graph give rise to bipartite partial duals. In this paper, we extend their result from plane graphs to ribbon graphs and obtain the first main result. Theorem 1.4. Let G be an ribbon graph and A ⊂ E(G). Then GA is bipartite if and only if A is the set of c-edges arising from ˆ an all-crossing direction of G m , the modified medial graph (which is defined in Section 2.2) of G. In [7], Huggett and Moffatt also found a sufficient condition for a subset of edges to give rise to an Eulerian partial dual in terms of all-crossing directions of medial graphs. In [10], Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. In this paper, we find more simple directions, i.e. crossing-total directions, of medial graph to characterize all Eulerian partial duals of any ribbon graph and obtain our second main result. Theorem 1.5. Let G be a ribbon graph and A ⊂ E(G). Then GA is Eulerian if and only if A is union of the set of all d-edges and the set of some t-edges arising from a crossing-total direction of Gm . 2. Preliminaries In this section we only give necessary preliminaries and refer the readers to the monograph [6] for details on ribbon graphs, their equivalent notions such as arrow presentations, and their partial duals. 2.1. Three types of line segments Definition 2.1. A ribbon graph G = (V (G), E(G)) is a (possibly nonorientable) surface with boundary represented as the union of two sets of topological discs: a set V (G) of vertices and a set E(G) of edges such that:
• the vertices and edges intersect in disjoint line segments; • each such line segment lies on the boundary of precisely one vertex and precisely one edge; • every edge contains exactly two such line segments. Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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We shall call these line segments in Definition 2.1 common line segments as in [9] since they belong to boundaries of both vertex discs and edge discs. Let G be (an abstract or a ribbon) graph and v ∈ V (G). We denote by dG (v ) the degree of v in G, i.e. the number of half-edges incident with v . Definition 2.2 ([9]). Let G be a ribbon graph, v ∈ V (G) and e ∈ E(G). By deleting the common line segments from the boundary of v , we obtain dG (v ) disjoint line segments, we shall call them vertex line segments. By deleting common line segments from the boundary of e, we obtain two disjoint line segments, we shall call them edge line segments. On a boundary component of a ribbon graph, vertex line segments and edge line segments alternate. The number of vertex line segments (equivalently, edge line segments) on a boundary component of a ribbon graph is called the degree of the boundary component. As an example, in Fig. 2(a), all vij ’s are vertex line segments, all eij ’s are edge line segments and all e′ij ’s are common line segments. v11 e32 v23 e21 v31 e12 v22 e31 v12 e52 is a boundary component of degree 5. The following lemma is no more than an observation. Lemma 2.3.
Let G be a ribbon graph and A ⊂ E(G). Let GA be the partial dual of G with respect to A. Then
(1) there is a natural bijection between vertex line segments of G and GA ; common line segments and edge line segments of edges inside A exchange after taking the dual; and common line segments and edge line segments of edges outside A keep unchanged after taking the dual. (2) the boundary components of G consist of vertex line segments and edge line segments; the boundary component of GA consist of vertex line segments, common line segments of edges inside A and edge line segments of edges outside A. (3) the boundary components of vertex discs of G consist of vertex line segments and common line segments; the boundary components of vertex discs of GA (i.e. circles of the arrow presentation of GA ) consist of vertex line segments, common line segments of edges outside A and edge line segments of edges inside A. (4) there is a natural bijection between boundary components of G and boundary components of (V (GA ), Ac ). Proof. We use Fig. 2 to show the lemma. For (1), see (a) and (d). (b) and (c) are intermediate steps from (a) to (d). For (2), see (a) and (e). For (3), see (a) and (d) again. For (4), see (a) and (f). □ As shown in Fig. 2(e) and (h), there is a bijection between boundary components of GA and vertex boundaries of GA and Observation 1.2 is thus established.
c
2.2. Medial graphs and directions Let G be a cellularly embedded graph in Σ . We construct its medial graph Gm by placing a vertex v (e) on each edge e of G and then for each face f with boundary v1 e1 v2 e2 · · · vd(f ) ed(f ) , drawing the edges (v (e1 ), v (e2 )), (v (e2 ), v (e3 )), · · ·, (v (ed(f ) ), v (e1 )) non-intersected in a natural way inside the face f . Note that Gm is also a cellularly embedded graph in Σ and a 4-regular one. We can color the faces of the medial graph Gm containing a vertex of G black and the remaining faces white, obtaining the canonical checkerboard colouring of Gm . See Fig. 3(a) and (b) for an example. A crossing-total direction of Gm is an assignment of an orientation to each edge of Gm in such a way that for each vertex v (e) of Gm , edges incident with v (e) are ‘‘in,in,out,out", ‘‘in,in,in,in" or ‘‘out,out,out,out" oriented in cyclic order with respect to v (e). See Fig. 3(c) for an example. If Gm is equipped with the canonical checkerboard colouring and a fixed crossing-total direction, then we can partition the vertices of Gm into three classes, called c-vertices, d-vertices and t-vertices, respectively according to the scheme shown in Fig. 4. Note that in Fig. 4, we draw the medial graph Gm of a ribbon graph G as a cellularly embedded graph, not as a ribbon graph. There is a natural bijection e ↔ v (e) between edges of G and vertices of Gm . The corresponding edges of G are called c-edges, d-edges and t-edges, respectively. See Fig. 3(d) for an example. Note that Gm = (G∗ )m and the canonical checkerboard colouring of (G∗ )m can be obtained from that of Gm by switching colours black and white. Lemma 2.4.
Let G be a cellularly embedded graph and e ∈ E(G). Under a fixed crossing-total direction of Gm , we have:
(1) e is a c-edge in G if and only if e is a d-edge in G∗ ; (2) e is a t-edge in G if and only if e is a t-edge in G∗ . In particular, a bridge e of a cellularly embedded graph G is always a d-edge in any fixed crossing-total direction of Gm . In [7], Huggett and Moffatt introduced the notion of all-crossing direction of Gm and then characterized all bipartite partial duals of a plane graph using all-crossing directions. An all-crossing direction is a crossing-total direction without t-vertices. Note that we can draw an orientable ribbon graph in the plane without twist in each vertex disc and each edge disc, and draw a non-orientable ribbon graph in the plane without twist in each vertex disc and with at most one twist in each edge disc. That is, for a given choice of orientation (clockwise direction) for each vertex disc, there are two types of edges Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 2. Illustration of Lemma 2.3: (a) a ribbon graph G and its common line segments, vertex line segments and edge line segments, (b) to (d) the construction of GA with A = {1, 2, 5} and corresponding line segments, (e) the boundary component of GA , (f) the boundary component of (V (GA ), Ac ), c (g) the marked spanning subgraph (V (G), Ac ) and (h) the arrow presentation of GA .
(twisted or untwisted) in a non-orientable ribbon graph. When forming the medial graph Gm , we must orient the vertex disk for each v (e) ∈ V (Gm ). If e is an untwisted edge of G with ends w , x we orient the disc for v (e) in the way that agrees with the orientations for the discs of w and x in G. If e is a twisted edge we choose an orientation for the disc for v (e); it will agree with the orientation of one end of e, say w , and disagree with the orientation of x. We must then apply twists to the two edges of the medial graph incident with v (e) that correspond to x. Some edges of the medial graph may be given two twists that cancel out. Thus, if there are τ twisted edges in G, then there are at most 2τ twisted edges in Gm . We add a 2-valent vertex (denoted by short slash) to each twisted edge in Gm , and denote the new graph by ˆ Gm and call it the modified medial graph. See Fig. 5(b). If Gm is equipped with the canonical checkerboard colouring, then we can obtain the checkerboard colouring of ˆ Gm naturally by neglecting extra artificial 2-valent vertices. If we direct edges of ˆ Gm such that every 4-valent vertex is c-vertex or d-vertex and every 2-valent vertex has ‘‘in, in’’ or ‘‘out, out’’ direction, we call the direction of ˆ Gm an all-crossing direction of ˆ Gm . See Fig. 5(c). When there are no such 2-valent vertices in ˆ Gm , ˆ Gm reduces to Gm . Fig. 5 gives an example of bipartite partial dual of G based on an all-crossing direction of ˆ Gm . Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 3. (a) a plane graph G, (b) its medial graph Gm with canonical checkerboard coloring, (c) a crossing-total direction of Gm , and (d) the corresponding {c , d, t }-edges of G.
Fig. 4. c-vertex, d-vertex and t-vertices.
Fig. 5. (a) a non-orientable ribbon graph G, (b) the modified medial graph ˆ Gm embedded in the ribbon graph G, (c) an all-crossing direction of ˆ Gm , and (d) arrow presentation of the bipartite partial dual G{5} .
2.3. Graph states Let Gm be a canonically checkerboard coloured medial graph and v (e) ∈ V (Gm ). There are two ways to split v (e) into two vertices of degree 2 as shown in Fig. 6. We call them white smoothing and black smoothing respectively. Let G be a cellularly embedded graph in Σ . A state S of G is a choice of white smoothing and black smoothing at each vertex of Gm . After splitting each vertex according to the state S, we obtain a set of disjoint closed cycles in Σ , call them state circles of S. We denote the number of state circles of S by c(S). A state is called even if each of its state circles is a cycle of even length. We denote the set of even states of G by E S . Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 6. White and black smoothings.
Fig. 7. Renaming vertex line segments for GA in Fig. 2.
Let A ⊂ E(G) and VA = {v (e)|e ∈ A} ⊂ V (Gm ). A state SA of G associated with A is the choice of choosing white smoothing for each vertex of VA and black smoothing for each vertex of V (Gm )\VA . Lemma 2.5. Let G ⊂ Σ be a cellularly embedded graph and A ⊂ E(G). Let SA be the graph state of G associated with A. Then there is a bijection between the vertices (of degree 2) of state circles of SA and the marking arrows (ignoring directions) of the arrow presentation of GA . Proof. Compare Fig. 6 with Definition of arrow presentation in [6]. □ Lemma 2.5 is a key observation that will be used in the proofs of the main results (Theorems 1.4 and 1.5). 3. Proofs Proof of Theorem 1.4. (H⇒) Let (X , Y ) be the bipartition of V (GA ). Suppose that X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl }. We rename each vertex line segment of GA as follows: if the vertex line segment vrs is on the boundary of xi , it will be renamed xi ; if it is on the boundary of yj , it will be renamed yj . An example is given in Fig. 7, where X = {x1 , x2 }, Y = {y1 , y2 }, ∂ (x1 ) ⊃ {v14 , v13 }, ∂ (y1 ) ⊃ {v11 , v12 }, ∂ (x2 ) ⊃ {v22 , v23 , v31 }, ∂ (y2 ) ⊃ {v21 , v33 , v32 }. See also Fig. 8(a), (a′ ), (a′′ ) and (a′′′ ). There is a natural bijection between vertex line segments of G and GA by Lemma 2.3 (1). We rename vertex line segments of G accordingly. Note that there is a natural bijection between vertex line segments of G and the edges of Gm . Thus we label each edge of Gm with X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl }. Then we label each edge of ˆ Gm with X = {x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yl } as Gm except that there are two identical labels for each twisted edge of Gm . In ˆ Gm , for an untwisted edge eh ∈ A, the label of edges incident to veh is shown as in Fig. 8(b); for an untwisted edge eh ∈ Ac , the label of edges incident to veh is shown as in Fig. 8 (b′ ); for a twisted edge eh ∈ A, the label of edges incident to veh is shown as in Fig. 8 (b′′ ); for a twisted edge eh ∈ Ac , the label of edges incident to veh is shown as in Fig. 8 (b′′′ ). According to the canonical checkerboard colouring of ˆ Gm , we direct the edge labelled xi and yj of ˆ Gm such that the black colour is on their right and left (using the orientation at the appropriate vertex of G to define right and left) along the directions, respectively, as shown in Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). It is obvious that the direction is an all-crossing direction of ˆ Gm . Furthermore each edge of A is a c-edge and each edge of Ac is a d-edge as shown in Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). (⇐H) Conversely, let ˆ Gm be equipped with the canonical checkerboard colouring. Let O be an all-crossing direction ˆ of Gm such that A is the set of c-edges arising from O. Then Ac is the set of d-edges. We label each edge of ˆ Gm with x if the black colour is on the right as we follow the direction of the edge, and with y if the black colour is on the left. See Fig. 8(c), (c′ ), (c′′ ) and (c′′′ ). Note that we can label x or y the twisted edge of ˆ Gm consistently. Let SA be the state of G associated with A. State circles of SA correspond to circles of arrow presentation of GA as shown in Fig. 8(a), (a′ ), (a′′ ) and Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 8. Proof of Theorem 1.4.
(a′′′ ). The circles labelled x and circles labelled y correspond to the circles in SA , i.e., the circles of the arrow presentation of GA . Then circles labelled x and circles labelled y form vertex bipartition of GA . Thus GA is bipartite. □ As a direct consequence, we have Corollary 3.1. Let G be a ribbon graph and A ⊂ E(G). Then GA is checkerboard colorable if and only if A is the set of d-edges arising from an all-crossing direction of ˆ Gm , the modified medial graph of G. Corollary 3.2. Let G be an orientable ribbon graph and A ⊂ E(G). Then GA is bipartite if and only if A is the set of c-edges arising from an all-crossing direction of Gm . Remark 3.3. Let G1 (left) and G2 (right) be non-orientable ribbon graphs shown in Fig. 9. They are dual to each other. (G1 )m and (G2 )m both have only one straight-ahead walks, the all-crossing direction of (G1 )m has no c-vertices and the all-crossing direction of (G2 )m has no d-vertices. This shows that Corollary 3.2 does not hold for non-orientable ribbon graphs. Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.
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Fig. 9. Examples which show that Corollary 3.2 does not hold for non-orientable ribbon graphs.
We now prove our second main but simpler result, a characterization of Eulerian partial duals of a ribbon graph in terms of crossing-total directions of its medial graph. But we first need the following crucial lemma. Lemma 3.4.
Let G be a ribbon graph and A ⊂ E(G). Then GA is Eulerian if and only if SA of G associated with A is even.
Proof. The degree of a vertex of GA is equal to the number of marking arrows of the corresponding circle of arrow presentation of GA . By Lemma 2.5, it is exactly the length of corresponding state circle of SA of G. Thus GA is Eulerian if and only if SA is even. □ Proof of Theorem 1.5. (H⇒) By Lemma 3.4, SA is even. So we can give an alternating (clockwise and counter-clockwise) orientation to each circle of SA . Transferring them to Gm , we obtain a crossing-total direction of Gm . It is easy to see that each edge of A is either a d-edge or a t-edge and each edge of Ac is either a c-edge or a t-edge. Thus A is the union of the set of all d-edges and some t-edges arising from a crossing-total direction of Gm . (⇐H) Conversely, if A is the union of the set of all d-edges and some t-edges arising from a crossing-total direction of Gm . Then SA is an even state of G since the edge orientations of each state circle of SA are alternating between clockwise and counterclockwise orientation. Thus GA is Eulerian by Lemma 3.4. □ Using Observation 1.2 and Lemma 2.4, we obtain: Corollary 3.5. Let G be a ribbon graph and A ⊂ E(G). Then GA is even-face graph if and only if A is the union of the set of all c-edges and the set of some t-edges arising from a crossing-total direction of Gm . Declaration of competing interest The authors declare that there is no conflict of interest in this paper. Acknowledgments This work is supported by NSFC, PR China (No. 11671336) and the Fundamental Research Funds for the Central Universities, PR China (No. 20720190062). We thank the referees and editor for helpful suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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Please cite this article as: Q. Deng, X. Jin and M. Metsidik, Characterizations of bipartite and Eulerian partial duals of ribbon graphs, Discrete Mathematics (2019) 111637, https://doi.org/10.1016/j.disc.2019.111637.