On homomorphisms of oriented graphs with respect to the push operation

Discrete Mathematics (

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On homomorphisms of oriented graphs with respect to the push operation Sagnik Sen Indian Statistical Institute, Bangalore, India

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Article history: Received 14 May 2015 Received in revised form 25 October 2016 Accepted 31 October 2016 Available online xxxx

An oriented graph G is a directed graph without directed cycles of length at most 2 having − → − → set of vertices V ( G ) and set of arcs A( G ). To push a vertex of an oriented graph is to reverse

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the orientation of the arcs incident to that vertex. If G′ can be obtained by pushing a set of

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− → − →′

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vertices of G , then we say G is in a push relation with G′ . A mapping f : V ( G ) → V ( H )

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is a pushable homomorphism of G to H if there exists a G which is in a push relation with

Keywords: Oriented graphs Push operation Graph homomorphism Chromatic number Planar graphs

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G such that uv ∈ A( G′ ) implies f (u)f (v ) ∈ A( H ). Klostermeyer and MacGillivray (2004) introduced pushable homomorphism and defined the pushable chromatic number of an − → − → − → oriented graph G as the minimum cardinality of V ( H ) such that G admits a pushable homomorphism to an oriented graph H. In this article, we further study the same topic and answer some of the questions asked in the above mentioned work, including studies of pushable chromatic numbers for the family of outerplanar graphs with girth restrictions, cactus, planar graphs and planar graphs with girth at least eight. © 2016 Elsevier B.V. All rights reserved.

1. Introduction

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An oriented graph G is a directed graph having no directed cycle of length 1 or 2 with set of vertices V ( G ) and set of arcs − → − → A( G ). We denote by G the underlying graph of G and G is an orientation of G. − → − → − → − → − → − → A homomorphism of G to H is a mapping φ : V ( G ) → V ( H ) such that uv ∈ A( G ) implies φ (u)φ (v ) ∈ A( H ). We write − → − → − → − → − → − → G → H whenever there exists a homomorphism of G to H . The oriented chromatic number χo ( G ) of G is the minimum − → − → − → order (number of vertices) of an oriented graph H such that G → H . − → To push a vertex v of a directed graph G is to change the orientations (that is, to replace an arc xy by yx) of all the arcs

− →

− →1

incident to v . Two oriented graphs G

− →1

− →2

and G

− →2

are in a push relation if it is possible to obtain G

by pushing some vertices

− →

of G . Note that the push relation is an equivalence relation. The class of oriented graphs that are in push relation with G − → is denoted by [ G ]. Note that the graphs from the same equivalence class have the same underlying graph. In fact, the push relation partitions

− →1 − →2

the set of all orientations of a simple graph G. That is why we are using notations such as G , G where 1, 2 are superscripts of the overhead arrow of G, and not G itself, referring to the fact that they are two different orientations of the same underlying simple graph G.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2016.10.023 0012-365X/© 2016 Elsevier B.V. All rights reserved.

Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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Fig. 1. Some oriented graphs.

− → − → − → − → − →1 − → ∈ [ G ] such that φ − → − → − → push − → − → − → is a homomorphism of G to H . We write G −−→ H whenever there exists a pushable homomorphism of G to H . The − → − → − → − → push − → pushable chromatic number χp ( G ) of G is the minimum order of an oriented graph H such that G −−→ H . A pushable homomorphism of G to H is a mapping φ : V ( G ) → V ( H ) such that there exists G 1

A (pushable) isomorphism is a bijective (pushable) homomorphism whose inverse is also a (pushable) homomorphism. Two oriented graphs are (pushably) isomorphic if there exists a (pushable) isomorphism between them. The oriented (or − → − → pushable) chromatic number χo (G) (or χp (G)) of an undirected graph G is the maximum χo ( G ) (or χp ( G )) taken over all

− →

orientations G of G. The oriented (or pushable) chromatic number χo (F ) (or χp (F )) of a family F of graphs is the maximum of the oriented (or pushable) chromatic numbers of the graphs from the family F . − → − → − → Let G be an oriented graph with vertex set V (G) = {v1 , v2 , . . . , vk }. The anti-twinned graph P( G ) of G is the oriented − → − → graph with set of vertices V (P( G )) = {v1 , v2 , . . . , vk } ∪ {v1′ , v2′ , . . . , vk′ } and set of arcs A(P( G )) = {vi vj , vi′ vj′ , vj vi′ , vj′ vi |

− → vi vj ∈ A( G )}. An oriented graph is anti-twinned if it can be expressed as an anti-twinned graph of some oriented graph.

Oriented coloring (see survey [21] for details) and vertex pushing (see [4,7,8,10,16–18] for details) are well-studied topics. Ochem and Pinlou [14] used the push operation on oriented graphs for proving upper bounds of the oriented chromatic number for the families of triangle-free planar graphs and 2-outerplanar graphs, respectively. Klostermeyer and MacGillivray [9] defined and studied the pushable chromatic number of oriented graphs and showed that given an oriented − → − → graph G it is NP-hard to decide whether χp ( G ) ≤ k for any k ≥ 5.

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Klostermeyer and MacGillivray also showed that if G is pushably isomorphic to H then P( G ) is isomorphic to P( H ) and asked if the converse is true. Brewster and Graves [3] showed that a similar question has a positive answer for edge-colored graphs and observed that Klostermeyer and MacGillivray’s question can be answered using the same technique, restricted to edge-colored graphs with multiplicity 2 [3]. Here we prove the converse of Klostermeyer and MacGillivray’s [9] result using a different, straight-forward technique. We also extend the theory following suggestions of Klostermeyer and MacGillivray [9] by studying the pushable chromatic number of the families of outerplanar graphs and planar graphs with girth restrictions, respectively, where girth is the length of a smallest cycle. In Section 2 we fix the notations and terminologies. In Section 3 we study the properties of pushable homomorphisms and their relation with homomorphisms of oriented graphs. In Sections 4 and 5 we study the pushable chromatic number for the families of outerplanar and planar graphs, respectively. Finally, in Section 6 we conclude our work and propose some future directions of research on this topic. 2. Preliminaries and some basic observations

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Let G be an oriented graph. For an arc uv of G the vertex u is an in-neighbor or −-neighbor of v and v is an out-neighbor or +-neighbor of u. The set of all in-neighbors and the set of all out-neighbors of v are denoted by N − (v ) and N + (v ), respectively. The set of all neighbors of v is N(v ) = N − (v ) ∪ N + (v ). The degree d(v ) of a vertex v is the cardinality |N(v )| of the set of neighbors of v . Similarly, in-degree and out-degree of a vertex v are given by d− (v ) = |N − (v )| and d+ (v ) = |N + (v )|, respectively. − → Given a set of vertices S of G define the set N α (S) = {u ∈ N α (v )|v ∈ S } for each α ∈ {+, −}. Furthermore, we will also αβ α use the notation N (v ) = N (N β (v )) for α, β ∈ {+, −}. We will use the four graphs depicted in Fig. 1 and the following observations to prove some of the results in this article.

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Observation 2.1. The directed 3-cycle C3 is the unique tournament on three vertices up to pushable isomorphism.

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Proof. There are only two tournaments on 3 vertices, namely the directed tournament C3 and the transitive tournament, − → which can be obtained by pushing any vertex of C3 . □

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Observation 2.2. The pushable chromatic number of UC 4 is χp (UC 4 ) = 4. Thus if any two vertices of an oriented graph form

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part of a UC 4 , then they must have different images under any pushable homomorphism. Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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Observation 2.3. Let P = v1 v2 . . . vk be an oriented path for some k ≥ 5. For any function f : {v1 , vk } → V ( C3 ) with

− →′

− →

f (v1 ) ̸ = f (vk ), there is an orientation P of P obtained by pushing vertices of V ( P ) \ {v1 , vk } such that f can be extended to a

− →

− →

homomorphism fext : P ′ → C3 . Proof. First we will show that the observation holds for k = 5. Note that the path P = v1 v2 . . . v5 has four edges and thus − → can have 24 = 16 different kinds of orientations. Suppose P is an orientation of P. Note that the number of forward arcs − → (arcs of the type vi vi+1 ) and the number of backward arcs (arcs of the type vi+1 vi ) in P have the same parity. If both these − → numbers are even, then we call P an even orientation of P and an odd orientation of P, otherwise. Thus there are 8 even − → orientations and 8 odd orientations of P . − → − → Let S( P ) denote the set of all orientations of P obtained from P by pushing vertices from the set {v2 , v3 , v4 }. Note that − → − → if P is an even (or odd) orientation, then all the elements of S( P ) are even (or odd) orientations. Also as there are 8 distinct − → − → − → subsets of the set {v2 , v3 , v4 }, the set S( P ) has 8 elements. Thus S( P ) is the set of all even (or odd) orientations of P, if P is an even (or odd) orientation of P. − → First assume that P is an even orientation of P. If f (v5 ) ∈ N + (f (v1 )), then we can use the orientation of P with four forward arcs. If f (v5 ) ∈ N − (f (v1 )), then we can use the orientation of P with four backward arcs. − → Next assume that P is an odd orientation of P. If f (v5 ) ∈ N + (f (v1 )), then we can use an orientation of P with three backward arcs. If f (v5 ) ∈ N − (f (v1 )), then we can use an orientation of P with three forward arcs. This proves the case when k = 5. − → Assume that k > 5 and the observation holds for all n ≤ k − 1. Let P = v1 v2 . . . vk be an oriented path and let − → f : {v1 , vk } → V ( C3 ) be a function with f (v1 ) ̸ = f (vk ). Let α be such that vk−1 ∈ N α (vk ). Now consider the α -neighbor of f (vk ). If it is not equal to f (v1 ), then we are done using induction. Otherwise, push vk−1 to arrive at the above situation. □

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Observation 2.4. Let C = v1 v2 . . . vk v1 be an oriented cycle for some k ̸ = 4, k ≥ 3. Given a fixed vertex x ∈ V ( C3 ), there exists

− →′

− →

− →′

an orientation C of C obtained by pushing some of the vertices of V ( C ) \ {v1 } such that f : C

− →

→ C3 with f (v1 ) = x.

Proof. The statement is clear for k = 3 by Observation 2.1. For k ≥ 5, let f (v1 ) = x. Let α be such that vk ∈ N α (v1 ). − → Furthermore, assume y ∈ N α (x) in C3 . Then fix f (vk ) = y. Now use Observation 2.3 to complete the proof as v1 v2 . . . vk is an oriented path with k ≥ 5 and f (v1 ) = x ̸ = y = f (vk ). □

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Observation 2.5. Given any vertex v ∈ V ( C

3+ ) we have N

++

− →

(v ) ∪ N −− (v ) = V ( C

3+ )

− → \ {v} and N +− (v ) ∪ N −+ (v ) = V ( C 3+ ).

A subgraph H of G is denoted by H ⊆ G. 3. Properties of pushable homomorphism First we will prove a result that will give us more insight regarding the relation between oriented graph homomorphisms and push operations.

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− →1

Proposition 3.1. Let f be a homomorphism of G to H . Then for each H

− →1

homomorphism of G

− →1

− → − →1 − → ∈ [ H ] there exists a G ∈ [ G ] such that f is a

to H .

− → − → − →1 − → − →1 − → ∈ [ H ]. Suppose one can obtain H from H by pushing the set of 1 − → − → − → − → vertices V1 ⊆ V ( H ). Now obtain G from G by pushing the pre-images of V1 , that is, the set of vertices f −1 (V1 ) ⊆ V ( G ). − →1 − →1 Proof. Let f be a homomorphism of G to H and let H

Clearly f is a homomorphism of G

to H .

□

The following observation that directly follows from the definition of anti-twinned oriented graphs will be instrumental in proving other results of this article.

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Observation 3.2. An oriented graph S is anti-twinned if and only if it is possible to partition the set of vertices V ( S ) into two parts V1 and V2 with a bijection f : V1 → V2 such that for all u ∈ V1 we have N + (u) = N − (f (u)) and N − (u) = N + (f (u)). Next we will prove a useful lemma.

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− →

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push

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Lemma 3.3. For any anti-twinned graph P( T ) we have G → P( T ) if and only if G −−→ T .

− → − →

− → − →

− → push − → − → all x ∈ V ( T )). Then G −−→ T by composition.

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push

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− →

Proof. If G → P( T ), then G −−→ P( T ). Note that ψ : P( T ) −−→ T where ψ (x) = ψ (x′ ) = x for x ∈ V ( T ) (push x′ for push

Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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push

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)

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If G −−→ T , then P( G ) → P( T ) due to Klostermeyer and MacGillivray [9]. Also G → P( G ) by inclusion. Thus we are done by composing the homomorphisms as the composition of pushable homomorphisms is a pushable homomorphism by Proposition 3.1. □ We are going to prove the converse of Theorem 3(b) from the article of Klostermeyer and MacGillivray [9].

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Theorem 3.4. An oriented graph G is pushably isomorphic to H if and only if P( G ) is isomorphic to P( H ). Proof. Klostermeyer and MacGillivray [9] proved the ‘‘only if’’ part and thus we are going to prove the ‘‘if’’ part of the theorem. − → − → If x is a vertex of an oriented graph X then x′ denotes its corresponding anti-twin in P( X ). Moreover, we fix the − → − → − → convention (x′ )′ = x. For any isomorphism f of P( G ) to P( H ) define the set Yf = {v ∈ V ( G )|f (v ′ ) = f (v )′ }.

− →

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− →

We proceed by induction on |V ( G ) \ Yf |. First we will deal with the base case |V ( G ) \ Yf | = 0, that is, Yf = V ( G ).

− → − → − → − → − → − → − → − → − → P( H )[V ( G )] denote the oriented graph induced by f (V ( G )) from P( H ). If we push the set of vertices V ( H ) \ f (V ( G )) of − → − → − → − → − → − → → be the P( H )[V ( G )], we obtain H according to the definition of anti-twinned graphs. Thus, P( H )[V ( G )] ∈ [ H ]. Let fV (− G) − → − → − → − → → : G → P( H )[V ( G )] is also an isomorphism. Therefore, restriction of the function f to V ( G ). As f is an isomorphism, fV (− G) − → push − → → : G − f V (− − → H is a pushable isomorphism. G) − → − → − → − → Then let |V ( G ) \ Yf | = k for some 1 ≤ k ≤ |V ( G )|. Suppose G is pushably isomorphic to H if there exists an − → − → − → isomorphism h of P( G ) to P( H ) such that |V ( G ) \ Yh | < k. − → Thus we can assume that there exists a v ∈ V ( G ) \ Yf , that is, f (v ′ ) ̸ = f (v )′ . Now define the following: ⎧ ⎨f (x) if x ̸ = v ′ and x ̸ = f −1 (f (v )′ ), ′ g(x) = f (v ) if x = f −1 (f (v )′ ), ⎩ ′ f (v ) if x = v ′ .

If Yf = V ( G ), then given any vertex x of H the image f (V ( G )) either contains x or its anti-twin x′ , but not both. Let

Let f −1 (f (v )′ ) = u. As f is a bijection and we just interchanged the images of v ′ and u = f −1 (f (v )′ ) to obtain g from f , the − → − → function g is also a bijection from V (P( G )) to V (P( H )). − → − → First we will show that g is a homomorphism of P( G ) to P( H ). Note that f (v ) and f (v )′ are non-adjacent. Hence v ′ and ′ u are non-adjacent, that is, {a, b} ̸ = {v , u}. Similarly {f (a), f (b)} ̸ = {f (v ′ ), f (u)} = {f (v ′ ), f (v )′ }. − → Let ab be an arc in P( G ). If a, b ̸ ∈ {v ′ , u} then there is an arc from g(a) = f (a) to g(b) = f (b) as f is an isomorphism. Now suppose a = v ′ and b ̸ ∈ {v ′ , u}. Then b ∈ N + (v ′ ) = N − (v ). Therefore, g(b) = f (b) ∈ N − (f (v )) = N + (f (v )′ ) = N + (g(a)). Similarly, one can argue for the case when b = v ′ and a ̸ ∈ {v ′ , u}. Then suppose that a = u and b ̸ ∈ {v ′ , u}, that is, b ∈ N + (u). Hence g(b) = f (b) ∈ N + (f (v )′ ) = N − (f (v )) = N + (f (v ′ )) = N + (g(f −1 (f (v )′ ))) = N + (g(a)). Similarly, one can argue for the case when b = u and a ̸ ∈ {v ′ , u}. − → Now we will show that g is an isomorphism by proving that if a and b are non-adjacent in P( G ), then g(a) and g(b) − → are non-adjacent in P( H ). If a, b ̸ ∈ {v ′ , u} are non-adjacent, then g(a) = f (a) and g(b) = f (b) are non-adjacent as f is an isomorphism. If a = v ′ and b ̸ ∈ {v ′ , u} are non-adjacent, then v and b are non-adjacent. This implies that f (v ) and f (b) = g(b) are non-adjacent. Thus f (v )′ = g(a) and g(b) are non-adjacent. If a = u and b ̸ ∈ {v ′ , u}, then v and b are non-adjacent. This implies that f (v ) and f (b) = g(b) are non-adjacent. Thus f (v ′ ) = g(a) and g(b) are non-adjacent. So we have shown that g is an isomorphism. Also note that v, v ′ ̸ ∈ Yf , Yf ⊆ Yg and v ∈ Yg . Hence Yf ⊊ Yg . Therefore,

− → |V ( G ) \ Yg | < k and we are done using induction. □ 4. Outerplanar graphs

An outerplanar graph is a graph that can be drawn in the plane with all its vertices lying on a circle and all its edges can be drawn inside the circle without any crossing. A cactus graph is a graph in which each edge belongs to at most one cycle. Let Ok denote the family of outerplanar graphs with girth at least k. Klostermeyer and MacGillivray [9] showed that χp (O3 ) = 4 and asked which subclasses of O3 have pushable chromatic number at most 3. We partially answer the question. In fact, the next result determines the exact values of the pushable chromatic number for the family of outerplanar graphs with girth at least k for all k ≥ 3. Theorem 4.1. For the family Ok of outerplanar graphs with girth at least k we have (a) χp (Ok ) = 4 for k = 3, 4, (b) χp (O3 ) = 3 for k ≥ 5. Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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Proof. (a) For k = 3, 4 the upper bound is due to Klostermeyer and MacGillivray [9] while the lower bound follows from − → Observation 2.2, that is, χp (UC 4 ) = 4. (b) For k ≥ 5 the lower bound follows from the fact that any cycle of odd length at least 5 has pushable chromatic number at least 3. − → For proving the upper bound suppose that M be a minimal (with respect to subgraph inclusion) oriented outerplanar − → − → graph with girth at least 5 that does not admit a pushable homomorphism to C3 . If M contains a vertex u of degree 1, − → − → − → then M \ {u} admits a pushable homomorphism to C3 . Since every vertex of C3 has in-degree and out-degree equal to − → − → − → − → 1, the pushable homomorphism of M \ {u} to C 3 can be extended to obtain a pushable homomorphism of M to C3 , a − → contradiction. Thus M has minimum degree at least 2. − → Hence M contains a face ux1 x2 . . . xl−2 v of length l ≥ 5 with at least (l − 2) consecutive vertices x1 , x2 , . . . , xl−2 of

− →

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push

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degree 2 [15]. Due to the minimality of M there exists a pushable homomorphism f : M \ {x1 x2 . . . xl−2 } −−→ C3 . Note that − → − → f (u) ̸ = f (v ) as u and v are adjacent. Thus f can be extended to a pushable homomorphism of M to C3 by using Observation 2.3 on the oriented path ux1 x2 . . . xl−2 v , a contradiction. □ Let C4 denote the undirected 4-cycle. As χp (C4 ) = 4 by Observation 2.2, any undirected simple graph with pushable chromatic number at most 3 must not contain C4 as a subgraph. In fact, if we restrict ourselves to the family C of cactus graphs, then having no C4 as subgraph is a necessary and sufficient condition for a cactus graph to have pushable chromatic number at most 3. Theorem 4.2. For the family C of cactus graphs χp (C ) = 4. Moreover, an undirected cactus graph F has χp (F ) ≤ 3 if and only if it does not contain a C4 as a subgraph. Proof. Note that C is a subfamily of O3 and C4 is a cactus. Thus, χp (C ) = 4 by Theorem 4.1(a).

− →

− →

Let M be a minimal (with respect to subgraph inclusion) oriented cactus, not containing C4 as a subgraph, such that M − → − → does not admit a homomorphism to C3 . Similarly as in the proof of Theorem 4.1(b), we can show that M has minimum degree at least 2. Therefore, M contains a cycle X of length k (̸ =4) that has at most one vertex with degree greater than 2. Let

− →′

− → − →

M be the graph obtained by deleting the degree two vertices of the cycle X . Due to minimality of M there exists a pushable

− →′

push

− →

− →

homomorphism f : M −−→ C3 . Thus f can be extended to a pushable homomorphism of M to C3 by Observation 2.4, a contradiction. □ Note that the above result cannot be extended to the family of outerplanar graphs as the following example indicates. Note − → − → that the oriented outerplanar graph D , depicted in Fig. 1, does not contain C4 as a subgraph. We will show that χp ( D ) = 4.

− →

− →

− → − → − → Now we will show that each element of [ D ] contains either a directed 5-cycle, or a transitive 3-cycle. Note that in Fig. 1 − → the oriented graph D contains a directed 5-cycle a1 a2 a3 a4 a5 a1 . Klostermeyer and MacGillivray [9] showed that the graph − → − → obtained by pushing a set S of vertices of an oriented graph G is the same as the graph obtained by pushing the set V ( G ) \ S − → − → of vertices of G . Thus it is possible to obtain any element of [ D ] by pushing at most two vertices from A = {a1 , a2 , a3 , a4 , a5 }.

As D contains K3 , χp ( D ) ≥ 3. If χp ( D ) = 3, then D must admit a pushable homomorphism to C3 by Observation 2.1.

If we push 0 vertices of A, then the so obtained graph will have the directed 5-cycle a1 a2 a3 a4 a5 a1 . If we push 1 or 2 vertices of A, then without loss of generality we can assume that either a1 or a1 , a2 or a1 , a3 are pushed. Note that this will force a transitive 3-cycles induced by {a1 , a5 , a51 } irrespective of which vertices from {a12 , a23 , a34 , a45 , a51 } are pushed. − → − → − → Thus none of the elements of [ D ] admit an oriented homomorphism to C3 . Therefore, χp ( D ) = 4. 5. Planar graphs A planar graph is a graph that can be drawn in the plane such that its edges do not cross. An undirected simple graph G admits an acyclic k-coloring if it can be colored by k colors in such a way that the graph induced by each color is an independent set and the graph induced by each pair of colors is a forest [5]. Theorem 5.1. Every graph with acyclic chromatic number at most k has pushable chromatic number at most k · 2k−2 .

− →

Proof. For any positive integer k the Zielonka graph [20] Zk of order k · 2k−1 is the oriented graph with set of vertices − → V ( Zk ) = ∪i=1,2,...,k Si where Si = {x = (x1 , . . . , xk )|xj ∈ {0, 1} for j ̸ = i and xi = ∗} and set of arcs A(Zk ) = {xy | x = (x1 , . . . , xk ) ∈ Si , y = (y1 , . . . , yk ) ∈ Sj and either xj = yi and i < j or xj ̸ = yi and i > j}. Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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− →

Furthermore, note that the vertices of a Z

k

)

–

can be partitioned into two disjoint sets of equal size

V1 = {x = (x1 , . . . , xk ) ∈ Si |x1 = 1 or x1 = ∗ and x2 = 1 for i = 1, 2, . . . , k}, V2 = {x = (x1 , . . . , xk ) ∈ Si |x1 = 0 or x1 = ∗ and x2 = 0 for i = 1, 2, . . . , k}. Also, we can define a function f : V1 → V2 by f ((x1 , . . . , xi−1 , ∗, xi+1 , . . . , xk )) = (x1 + 1, . . . , xi−1 + 1, ∗, xi+1 + 1, . . . , xk + 1)

− →

where the + operation is taken modulo 2. It is clear that f is a bijection of the type described in Observation 3.2. Hence Zk − → −−−→ −−−→ − → is an anti-twinned oriented graph. Therefore, Zk = P(Zk [V1 ]) where Zk [V1 ] is the oriented graph induced from Zk by V1 . − → Now let G be an oriented graph such that G admits an acyclic k-coloring. Raspaud and Sopena [19] showed that

− →

− →

−−−→ − → push −−−→ = P(Zk [V1 ]). Hence by Lemma 3.3 G −−→ Zk [V1 ]. □ − → − → − → As χp ([ G ]) ≤ χo ( G ) ≤ 2χp ([ G ]) [9], the upper bound of the above result is tight for k ≥ 3 due to Ochem [13]. Now we

G → Z

k

establish the lower and upper bounds for the pushable chromatic number of planar graphs. Theorem 5.2. Let P3 be the family of all planar graphs. Then 10 ≤ χp (P3 ) ≤ 40. Proof of the upper bound of Theorem 5.2. Borodin [1] showed that every planar graph admits an acyclic 5-coloring. Thus, the upper bound follows from Theorem 5.1. □ For convenience, the proof of the lower bound is broken into several smaller results in the following.

− →

Proposition 5.3. There exists an oriented graph H on χp (P3 ) vertices such that every oriented planar graph admits a pushable

− →

homomorphism to H .

− → − →

Proof. Let OGχp (P3 ) be the set of all oriented graphs of order χp (P3 ). If our claim is false, then for each G ∈ OGχp (P3 ) there

− →

exists an oriented planar graph P

− →

− → G

− →

that does not admit a homomorphism to G . Let X =

− →

⨆

− → G ∈OGχp (P ) 3

P

− → G

be the disjoint

union of all such graphs. Note that X is a planar graph and does not admit homomorphism to any oriented graph on χp (P3 )

− →

vertices and thus has χp ( X ) > χp (P3 ), a contradiction. □

− →

− →

− →

An oriented graph T is minimal universal pushable target for the family of planar graphs (MUTP) if |V ( T )| = χp (P3 ), push

− →

− →

− →

G −−→ T for all G ∈ P3 and T is minimal such graph with respect to subgraph inclusion. − → − → By Proposition 5.3 there exists an MUTP, say, T . Note that by Proposition 3.1 any element of [ T ] is also an MUTP. Moreover, we will prove that the graph obtained by reversing all the arcs of an MUTP is also an MUTP.

− →

− →

− →

Lemma 5.4. Let T be an MUTP. Then the oriented graph Tr obtained by reversing all the arcs of T is also an MUTP.

− →

− →

− →

− →

Proof. First note that |V ( T r )| = |V ( T )| = χp (P3 ). Next let G be any oriented planar graph and let G

− →

− →

obtained by reversing all the arcs of G . Note that G

− →

push

− →

− →

r

− →

push

− →

− →

r be push

the graph

− →

∈ P3 and thus, G r −−→ T . This implies G −−→ T r . Hence

G −−→ Tr for all G ∈ P3 . − → − → − → − → Suppose that Tr is not an MUTP. Thus there exists a proper subgraph T ′ r of Tr such that |V ( T ′ r )| = χp (P3 ) and

→ − → − → − → push − − → G −−→ T ′ r for all G ∈ P3 . Note that the graph T ′ , obtained by reversing all the arcs of T ′ r , is a proper subgraph of − → → − → push − − → T . Following the arguments of the first paragraph of the proof we obtain that |V ( T ′ )| = χp (P3 ) and G −−→ T ′ for all − → − → G ∈ P3 , contradicting the minimality of T . □ − → Two vertices u and v of G agree on a third vertex w if w ∈ N α (u) ∩ N α (v ) for some α ∈ {+, −}. Similarly, u and v disagree on w if w ∈ N α (u) ∩ N β (v ) for {α, β} = {+, −}. Let Au,v denote the set of vertices that u and v agree on and let Du,v denote the set of vertices that u and v disagree on. Note that given two vertices u and v of a fixed oriented graph the sets Au,v and Du,v remain unchanged under push operation unless exactly one of u, v is pushed. In this case, the two sets are interchanged. Therefore, the parameters Mu,v = max{|Au,v |, |Du,v |} and mu,v = min{|Au,v |, |Du,v |} are invariant under the push operation. − → − → − →1 Consider an MUTP graph T . Let ab be an arc of T . Now push all in-neighbors of a. If the so-obtained graph T has 1 − → |Aa,b | ≥ |Da,b |, then stop. Otherwise, push the vertex a and reverse all the arcs of T and stop. Call the graph obtained in − → − → the final step H and fix it for the rest of this article. Notice that H is an MUTP with N + (a) = N(a) and |Aa,b | ≥ |Da,b |. − →

push

− →

− →

Lemma 5.5. If f : G −−→ K and mx,y ≥ 1 in G , then f (x) ̸ = f (y). Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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⃗0 with χp (B⃗0 ) = 8. (b) The gadget graph B⃗1 . The thick arrow suggests that there are arcs from v to every other vertex Fig. 2. (a) An oriented planar graph B except x and y.

− →

Proof. The proof follows from the fact that x, y disagree on at least one vertex in any element of [ G ].

− →

□

− →

Now we will make use of the minimality of H and prove that either H has at least 10 vertices or it is a tournament on 9 vertices.

− →

− →

Lemma 5.6. Either | H | ≥ 10 or H is a tournament on 9 vertices.

− →

− → 0 such that for any pushable homomorphism − → push − → − → − → − →1 − → − → f : X 0 −−→ H there exists X 0 ∈ [ X 0 ] with the following property: f : X 0 → H and for each arc uv ∈ A( H ) there − →1 exists an arc xy ∈ A( X 0 ) such that f (x) = u and f (y) = v . − → − → − → Now we construct an oriented planar graph X1 by gluing a copy of the planar graph B0 (Fig. 2(a)) to each vertex of X0 − → − → − → by identifying the vertex with the vertex x8 of B0 . Then we glue the gadget graph B1 (Fig. 2(b)) to each vertex of X1 by − → − → − → identifying the vertex with the vertex v of B1 and obtain a new oriented graph X2 . Note that the gadget graph B1 is planar − → − → − → and thus X2 is also planar. After that we construct another oriented planar graph X3 by gluing a copy of the planar graph B0 − → − → to each arc of X2 by identifying the arc with the arc x8 x4 of B0 . − → − → push − → As X2 is a planar graph, there exists a pushable homomorphism f : X2 −−→ H . − → − → Note that each pair of non-adjacent vertices of the oriented planar graph B0 from Fig. 2 is part of a common UC 4 . So no − → − → two vertices of B0 can have the same image under f by Observation 2.2 and thus χp ( B0 ) = 8. Moreover, |Ax4 ,x8 |, |Dx4 ,x8 | ≥ 3 − → − → − → − → in B0 . Thus, for each B0 contained in X2 we have |Af (x4 ),f (x8 ) |, |Df (x4 ),f (x8 ) | ≥ 3 in H . − → − → The vertices v, x and y of B1 must have different images under f as any pair of them are part of an UC 4 . Also the vertices of the directed 5-cycle that is induced by the common neighbors of v and x must have different images under any pushable − → − → homomorphism. Hence Mf (v ),f (x) ≥ 4 in H . Similarly, we have Mf (v ),f (y) ≥ 4 in H . − → − → − → Therefore, each vertex in H has degree at least 7 and for each arc ab of H we have ma,b ≥ 3. Also for each vertex a of H − → there are at least two vertices c and d of H with Ma,c , Ma,d ≥ 4. − → Therefore, each vertex u of H has a neighbor v such that mu,v ≥ 3 and Mu,v ≥ 4. That is, u and v have at least 7 common − → − → Proof. Due to the minimality of H there exists an oriented planar graph X 1

neighbors. Thus H has at least 9 vertices and if H has exactly 9 vertices then it is a tournament. □

− →

Now for the rest of the proof we will assume that H is a tournament on 9 vertices and prove the theorem by contradicting this assumption. − → − → − → First recall that there is a vertex a of H such that N + (a) = V ( H ) \ {a}. Moreover, there is a vertex b of H such that + + + − + + + |Aa,b | ≥ |Da,b |. This implies |N (a) ∩ N (b)| = 4 and |N (a) ∩ N (b)| = 3. Let N (a) ∩ N (b) = A and N (a) ∩ N − (b) = B.

− →

− →

For any vertex x ∈ V ( H ) \ {a} we have ma,x ≥ 3. Thus we must have 3 ≤ d+ (x) ≤ 4 for all x ∈ V ( H ) \ {a}. Lemma 5.7. Each vertex x ∈ A has exactly 1 in-neighbor in B. Proof. Suppose a vertex x ∈ A does not have exactly 1 in-neighbor in B. Then one of the following cases hold: (i) Suppose x has no in-neighbors in B. That means B ⊆ N + (x). As 3 ≤ d+ (x) ≤ 4, there can be at most 1 out-neighbor of x in A. The set Ax,b contains the vertex a, out-neighbors of x in A and in-neighbors of x in B. Thus |Ax,b | ≤ 2, a contradiction. (ii) Suppose x has exactly 2 in-neighbors in B. As 3 ≤ d+ (x) ≤ 4, there must be at least 2 out-neighbors of x in A. Thus there is at most 1 in-neighbor of x in A. The set Dx,b contains in-neighbors of x in A and out-neighbors of x in B. Thus |Dx,b | ≤ 2, a contradiction. Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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(iii) Suppose x has exactly 3 in-neighbors in B. As 3 ≤ d+ (x) ≤ 4, there must be at least 3 out-neighbors of x in A. Thus there is no in-neighbor of x in A. The set Dx,b contains in-neighbors of x in A and out-neighbors of x in B. Thus |Dx,b | = 0, a contradiction. Thus we are done. □ Now we are ready to prove the main result. Proof of the lower bound of Theorem 5.2. For the lower bound, recall that |A| = 4 and |B| = 3. Thus, by Lemma 5.7 and the pigeonhole principle, there are two vertices x, y ∈ A with a common in-neighbor in B and the other two vertices of B are common out-neighbors of x and y. Thus Ax,y ⊇ B ∪ {a, b}. Therefore, Dx,y ⊆ A \ {x, y}. That implies |Dx,y | ≤ |A \ {x, y}| ≤ 2, which contradicts the fact that ma,b ≥ 3 for every ab ∈ A(H). □ Note that, improving the upper bound will improve the long standing upper bound of oriented chromatic number of planar graphs. Indeed our result uses the proof of the latter. Whereas our lower bound proof is independent of the lower bound proof for oriented chromatic number of planar graphs by Marshall [12]. Moreover, a lower bound of 9 for the pushable chromatic number of planar graphs can be achieved using Marshall’s result while we provide a better lower bound of 10 for the same. Even though our lower bound does not imply any improvement of Marshall’s lower bound of 18 for the oriented chromatic number of planar graphs, it does imply the following corollary. Corollary 5.8. There does not exist any anti-twinned oriented graph on 18 vertices to which every oriented planar graph admits a homomorphism. 2|E(G)|

The average degree of a graph G is ad(G) = |V (G)| and the maximum average degree of a graph G is mad(G) = maxH ⊆G Now we will prove the following result using discharging method. Theorem 5.9. If mad(G) <

8 , 3

2|E(H)| . |V (H)|

then χp (G) ≤ 4.

− →

Proof. First assume that H is a minimal (with respect to the number of vertices) oriented graph with maximum average − → degree less than 8/3 that does not admit a pushable homomorphism to C 3+ . − → Next we will prove the following forbidden configurations for H .

− →

(i) A vertex of degree one: The graph obtained by deleting the degree one vertex from H admits a pushable homomorphism − → − → − → to C 3+ due to minimality of H . That pushable homomorphism can be extended to a pushable homomorphism of H − → − → to C 3+ as each vertex of C 3+ has degree at least one. − → (ii) Two adjacent vertices of degree two: The graph obtained by deleting the two adjacent vertices of degree two from H − → − → admits a pushable homomorphism to C 3+ due to minimality of H . That pushable homomorphism can be extended − → − → to a pushable homomorphism of H to C 3+ by Observation 2.5. (iii) A vertex of degree three adjacent to two vertices of degree two: The graph obtained by deleting the degree three vertex − → − → − → and its two neighbors of degree two from H admits a pushable homomorphism to C 3+ due to minimality of H . That − → − → pushable homomorphism can be extended to a pushable homomorphism of H to C 3+ by Observation 2.5.

− →

Now consider a charge function ch : V ( H ) → R defined by ch(v ) = d(v ). Note that

− → − → |V ( H )|

2·|A( H )|

− → ch(v ) v∈V ( H )

∑

=

− → |V ( H )|

< 38 .

Let each vertex of degree at least 3 give 1/∑ 3 charge to each of with degree 2. This procedure will give rise to a ∑its neighbors → ch(v ) = − → ch∗ (v ). Now let us calculate the values of the updated new updated charge function ch∗ satisfying v∈V (− H) v∈V ( H )

− →

charge function ch∗ based on the degree of the vertices of H . (i) If d(v ) = 2, then by forbidden configuration (ii), both its neighbors have degree at least 3. Therefore, it receives exactly 2 · 1/3 = 2/3 charge, and thus ch∗ (v ) = 2 + 2/3 = 8/3. (ii) If d(v ) = 3, then by forbidden configuration (iii), v gives away at most 1/3 charge. Therefore, ch∗ (v ) ≥ 3 − 1/3 = 8/3. (iii) If d(v ) = k ≥ 4, then it gives away at most k · 1/3 = k/3 charge. Therefore, ch∗ (v ) ≥ k − k/3 = 2k/3 ≥ 8/3. This implies

− → − → |V ( H )|

2·|A( H )|

− → ch∗ (v ) v∈V ( H )

∑

=

− → |V ( H )|

≥ 83 , a contradiction. □

Now we will prove a tight bound for the pushable chromatic number for the family of planar graphs with girth at least 8. Theorem 5.10. Let P8 be the family of all planar graphs with girth at least 8. Then χp (P8 ) = 4. Proof. If χp (P8 ) ≤ 3, then we can prove, similar to Proposition 5.3, that there exists a tournament on 3 vertices to which every oriented planar graph with girth at least 8 admits a pushable homomorphism. According to Observation 2.1 the directed Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023

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− →

3-cycle C3 is the only tournament on three vertices up to pushable isomorphism. Therefore, any oriented planar graph with − → girth at least 8 must admit a homomorphism to C3 by Proposition 3.1. Consider a graph with an oriented cycle on 9 vertices {u1 , u2 , . . . , u9 } and a tenth vertex v . For each i ∈ {1, 2, . . . , 9}, we have two vertex disjoint paths Pi = v xi1 xi2 xi3 ui and Pi′ = v yi1 yi2 yi3 ui connecting v and ui . Each Pi has four forward arcs v xi1 , xi1 xi2 , xi2 xi3 , xi3 ui and each Pi′ has three forward arc and a backward arc v yi1 , yi1 yi2 , yi2 yi3 , ui yi3 . Let us call this so

− →

obtained oriented graph Y .

− →

− →

− →

Note that Y is an oriented planar graph with girth 8. Notice that for any Y ′ ∈ [ Y ] we will have one path, with either three forward arcs and one backward arc or with three backward arcs and one forward arc, connecting v to ui for each i ∈ {1, 2, . . . , 9}. Observe that the 4-path x0 x1 . . . x4 with three forward arcs and one backward arc does not admit a homomorphism with − → − → − → x0 and x4 mapped to the same vertex of C3 . Now let f be a pushable homomorphism of Y to C3 . Then, because of the above observation, f (v ) ̸ = f (ui ) for each i ∈ {1, 2, . . . , 9}. But we know that the 9-cycle has pushable chromatic number at least 3. − → − → That means f must be onto the vertices of C3 when restricted to the 9-cycle. Hence f (v ) ̸ ∈ V ( C3 ), a contradiction. This gives the lower bound. We know that the family of graphs with maximum average degree less than 38 contains the family P8 of planar graphs with girth at least 8 as a proper subfamily due to Borodin, Kostochka, Nešetřil, Raspaud and Sopena [2]. Thus the upper bound follows by Theorem 5.9. □ 6. Conclusions A graph is called a core graph if it does not admit a homomorphism to any of its proper subgraphs [6]. The (up to isomorphism) unique minimal [6] subgraph to which a graph admits a homomorphism is called its core. Marshall [11] first established the lower bound of 17 for the oriented chromatic number of planar graphs by showing that there exists no oriented graph on 16 vertices to which every planar graph admits a homomorphism. For proving this, first he showed that if there exists a minimal (with respect to subgraph inclusion) graph on 16 vertices to which every planar graph admits a − → homomorphism, then it must be isomorphic to the Tromp graph [11] T 16 on 16 vertices. Then he constructed an example − → of an oriented planar graph that does not admit a homomorphism to T 16 . After that Marshall [12] extended his result to prove that the only oriented graph on 17 vertices to which all planar graphs can admit a homomorphism is an oriented graph − → − → whose core is T 16 . An easy but significant observation is that the family of Tromp graphs, in particular T 16 , are anti-twinned graphs. So if one can show that the only possible oriented core graph on 18 vertices to which every planar graph admits a homomorphism is an anti-twinned oriented graph, then by Corollary 5.8 the lower bound for oriented chromatic number can be improved to 19. Question. Is it possible to extend Corollary 5.8 to prove that there does not exist any oriented graph on 18 vertices to which every oriented planar graph admits a homomorphism? Another interesting future direction of work is to characterize all outerplanar graphs with pushable chromatic number at most three. Acknowledgment I acknowledge the anonymous reviewers for their valuable suggestions. References [1] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (3) (1979) 211–236. [2] O.V. Borodin, A.V. Kostochka, J. Nešetřil, A. Raspaud, É. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1–3) (1999) 77–89. [3] R.C. Brewster, T. Graves, Edge-switching homomorphisms of edge-colored graphs, Discrete Math. 309 (18) (2009) 5540–5546. [4] D.C. Fisher, J. Ryan, Tournament games and positive tournaments, J. Graph Theory 19 (2) (1995) 217–236. [5] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (4) (1973) 390–408. [6] P. Hell, J. Nešetřil, Graphs and Homomorphisms, in: Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2004. [7] W.F. Klostermeyer, et al., Hamiltonicity and reversing arcs in digraphs, J. Graph Theory 28 (1) (1998) 13–30. [8] W.F. Klostermeyer, Pushing vertices and orienting edges, Ars Combin. 51 (1999) 65–76. [9] W.F. Klostermeyer, G. MacGillivray, Homomorphisms and oriented colorings of equivalence classes of oriented graphs, Discrete Math. 274 (1–3) (2004) 161–172. [10] G. MacGillivray, K.L.B. Wood, Re-orienting tournaments by pushing vertices, Ars Combin. 57 (2000) 33–47. [11] T.H. Marshall, Homomorphism bounds for oriented planar graphs, J. Graph Theory 55 (2007) 175–190. [12] T.H. Marshall, On oriented graphs with certain extension properties, Ars Combin. 120 (2015) 223–236. [13] P. Ochem, Negative results on acyclic improper colorings, in: Proc. European Conference on Combinatorics, Graph Theory and Applications, EuroComb 2005, DMTCS Proc. AE, 2005, pp. 357–362. [14] P. Ochem, A. Pinlou, Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs, Graphs Combin. (2013) 1–15. [15] A. Pinlou, É. Sopena, Oriented vertex and arc colorings of outerplanar graphs, Inform. Process. Lett. 100 (3) (2006) 97–104.

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Please cite this article in press as: S. Sen, On homomorphisms of oriented graphs with respect to the push operation, Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.023