2-Edge-Colored Chromatic Number of Grids is at Most 9

A 2-edge-colored graph is a pair (G,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G, \sigma )$$\end{document} where G is a graph, and σ:E(G)→{'+','-'}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma :E(G)\rightarrow \{\text {'}+\text {'},\text {'}-\text {'}\}$$\end{document} is a function which marks all edges with signs. A 2-edge-colored coloring of the 2-edge-colored graph (G,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G, \sigma )$$\end{document} is a homomorphism into a 2-edge-colored graph (H,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H, \delta )$$\end{document}. The 2-edge-colored chromatic number of the 2-edge-colored graph (G,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G, \sigma )$$\end{document} is the minimum order of H. In this paper we show that for every 2-dimensional grid (G,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G, \sigma )$$\end{document} there exists a homomorphism from (G,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G, \sigma )$$\end{document} into the 2-edge-colored Paley graph SP9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SP_9$$\end{document}. Hence, the 2-edge-colored chromatic number of the 2-edge-colored grids is at most 9. This improves the upper bound on this number obtained recently by Bensmail. Additionally, we show that 2-edge-colored chromatic number of the 2-edge-colored grids with 3 columns is at most 8.


Introduction
In the whole paper we will use standard graph theory notations. A 2-edge-colored graph is a pair ðG; rÞ where G is a undirected graph and r : EðGÞ ! f' þ '; ' À 'g. For the vertex v 2 VðGÞ by N À ðvÞ (resp. N þ ðvÞ) we denote the set of neighbors of v such that the edge to v has assignment '-' (resp. 'þ'). Similarly for a set S & VðGÞ, we define N À ðSÞ ¼ S v2S N À ðvÞ and N þ ðSÞ ¼ S v2S N þ ðvÞ. A 2-edge-colored coloring of a 2-edge-colored graph G is a proper coloring / of V(G) such that if there exist two edges fu; vg and fx; yg with /ðuÞ ¼ /ðxÞ and /ðvÞ ¼ /ðyÞ, then these two edges have the same sign. The 2-edge-colored chromatic number of the 2-edge-colored graph G, denoted by v 2 ðGÞ, is the minimum number of colors needed for a 2-edge-colored coloring.
Equivalently, the 2-edge-colored chromatic number v 2 ðG; rÞ of the 2-edgecolored graph ðG; rÞ is the minimum order of the graph ðH; kÞ such that ðG; rÞ admits a 2-edge-colored homomorphism (preserving edges and signs) to ðH; kÞ. Graph ðH; kÞ we call a target graph or coloring graph. The 2-edge-colored chromatic number v 2 ðGÞ of a graph G is defined as maximum over all 2-edgecolored of G. For a graph class F , we define the 2-edge-colored chromatic number v 2 ðF Þ as the maximum over 2-edge-colored chromatic number for any members of F . For convenience, in the rest of the paper we will omit the r and simply write G for a 2-edge-colored graph ðG; rÞ if there is no ambiguity.
In this paper we focus on 2-edge-colored chromatic number for class of 2-dimensional grids G. The grid is defined as the graph being the Cartesian product of two paths. Similarly, by G k we denoted the grid with k columns.
In Sect. 2 we define 2-edge-colored Paley graph SP q and focus on SP 9 as a target graph in 2-edge-colored homomorphisms. That graph was used in this context earlier, for example Montejano et al. [6] show that there exist 2-edge-colored homomorphism from every 2-edge-colored outerplanar graph to SP 9 . Some other application of that target graph we can find in [7].
In Sect. 3 we prove: Theorem 1 For every 2-edge-colored grid G there exists a 2-edge-colored homomorphism h : G ! SP 9 .
In Sect. 4 we focus on grids with 3 columns and improve upper bound given by Bensmail. More precisely we show that: Theorem 3 For every 2-edge-colored grid G with 3 columns there exists a 2-edgecolored homomorphism h : G ! SP 9 nf0g.
Let q be a prime power such that q 1ðmod 4Þ and F q be finite field of order q. The Paley graph P q is undirected graph with vertex set VðP q Þ ¼ F q and edge set EðP q Þ ¼ È fx; yg : y À x is a non-zero square in F q É . Notice that À1 is a square in F q . Hence, if x À y is a square, then y À x is a square.
The 2-edge-colored Paley graph SP q is 2-edge-colored graph ðK q ; rÞ, where K q is the complete graph on vertices F q and rðfx; ygÞ ¼ 0 þ 0 () y À x is a non-zero square in F q . In this paper we use 2-edge-colored Paley graph SP 9 build over the Galois' field GFð3 2 Þ. Elements of this field are the polynomials of degree 1 over GF(3) with multiplication modulo x 2 þ 1. Elements f0; 1; 2; x; 2xg are squares and fx þ 1; x þ 2; 2x þ 1; 2x þ 2g are non squares. The graph P 9 is presented on Fig. 1.

Lemma 5 [8]
SP q is isomorphic to the 2-edge-colored graph constructed by flipping signs of all edges.
The following properties of SP 9 are well know:

Lemma 6
1. The Paley graph SP 9 has 72 automorphisms. 2. For every non-zero square a 2 F 9 and every b 2 F 9 , the function h a;b ðyÞ ¼ ay þ b is an automorphism in SP 9 . 3. The reflection function r : SP 9 ! SP 9 , which maps the polynomial ux þ v to vx þ u is an automorphism in SP 9 . 4. Each automorphism h in SP 9 is either of the form h ¼ h a;b or is a composition h ¼ r h a;b for some a and b.

Lemma 7
The Paley graph SP 9 is vertex-transitive and edge-transitive. Moreover for every two induced subgraphs S 1 and S 2 with jS 1 j ¼ jS 2 j ¼ 3 and the same number of edges marked with '-', there is an automorphism of SP 9 which maps S 1 on S 2 .
We shall call 3-elements induced subgraph T & VðSP 9 Þ a triangle if all its edges are marked with the same sign. Furthermore, we shall say that set S & VðSP 9 Þ is triangle free if the subgraph of SP 9 induced by S does not contain a triangle.
Lemma 9 For every vertex v 2 VðSP 9 Þ, the sets N þ ðvÞ and N À ðvÞ are triangle free.
Proof By Lemma 5 and 7, it is sufficient to consider the triangle free set S ¼ f0; 1; xg. Then N þ ðSÞ ¼ SP 9 nf2x þ 2g and N À ðSÞ ¼ SP 9 nf0g. h 3 Proof of Theorem 1 Consider a path (u, v, w) with arbitrary signs on the edges fu; vg and fv; wg. Suppose that we have: an arbitrary 3-elements triangle free set S 1 & VðSP 9 Þ of colors available in u and an arbitrary color b 2 VðSP 9 Þ for the vertex w. Then there is a 3-elements triangle free set S 2 & VðSP 9 Þ available in v. More precisely: Lemma 11 Consider a path (u, v, w) with arbitrary signs on the edges fu; vg and fv; wg. For every 3-elements triangle free set S 1 & VðSP 9 Þ and every color b 2 SP 9 , there exists a 3-elements triangle free set S 2 & VðSP 9 Þ such that for each s 2 2 S 2 there exists s 1 2 S 1 and a coloring c : ðu; v; wÞ ! VðSP 9 Þ with cðuÞ ¼ s 1 , cðvÞ ¼ s 2 , cðwÞ ¼ b.
Proof We will prove the lemma in case when both edges of path (u, v, w) are marked with 'þ'. In any other case the proof is similar.. By Lemmas 10 and 8, jN þ ðS 1 Þj ¼ 8 and jN þ ðbÞj ¼ 4. Hence, there exists 3-elements set S 2 & N þ ðs 1 Þ \ N þ ðbÞ. By Lemma 9, the set S 2 is triangle free. h Proof of Theorem 1 We color the 2-edge-colored grid G row by row. It is easy to color first row by SP 9 (in fact, we can do it using only four colors, for example by N þ ð0Þ ¼ f1; 2; 2x; 2x þ 1g). Assume now that, for k [ 1, the first k À 1 rows of G have been already colored and we color k-th row.
Let us denote the vertices in the k À 1-th row by a 1 ; a 2 ; . . .; a n and the vertices in the k-th row by b 1 ; b 2 ; . . .; b n . By Lemma 8, the vertex b 1 can be colored by four possible colors. By Lemma 9, any three of these colors form a triangle free set. Let us denote by S 1 any of these sets. Now for each i ¼ 2; 3; . . .; n we define set S i as a result of applying Lemma 11 for the set S iÀ1 and the color hða i Þ. Now we can color vertices b 1 ; b 2 ; . . .; b n in reverse order. First we choose any color in S n for hðb n Þ. For hðb nÀ1 Þ we set the color in S nÀ1 such that the sign of the edge ðhðb nÀ1 Þ; hðb n ÞÞ in SP 9 equals to the sign of the edge ðb nÀ1 ; b n Þ in the grid G. Notice that for each s 2 S nÀ1 , the sign of the edge ða nÀ1 ; b nÀ1 Þ in the grid is equals to the sign of the edge ðs; hða nÀ1 ÞÞ in SP 9 . Consecutive vertices we color in the same way. h

Grids with 3 Columns
In the previous section, we show that every 2-edge-colored grid admits 2-edgecolored coloring with 9 colors. This bound can be improved when we consider grids with a small number of columns. Bensmail [2] shows that 2-edge-colored chromatic number of grids with two columns is equal to 5. From the same paper we know that 7 v 2 ðG 3 Þ 9. We can improve this upper bound by showing that there exists a homomorphism from every 2-edge-colored grids with 3 columns to SP À0 9 , see Fig. 2. SP À0 9 is construct from SP 9 by removing vertex 0. It is easy to see that this graph has 8 automorphisms. Namely, four rotations (f ðyÞ ¼ ay where a is a nonzero square) and four reflection. SP À0 9 is also isomorphic to the graph constructed by flipping signs of all edges.
Lemma 12 Consider a path (u, v, w) with arbitrary signs on the edges fu; vg and fv; wg. For every color a 2 SP À0 9 and every b 6 2 fa; Àag there exists a coloring of path c : ðu; v; wÞ ! VðSP À0 9 Þ such that cðuÞ ¼ a and cðwÞ ¼ b.
Proof By automorphisms of SP À0 9 and the fact that it is isomorphic to the graph constructed by flipping signs of all edges, we can only consider the case when a ¼ 1.
This observation leads to partition of SP À0 9 into four sets: f1; 2g, fx; 2xg, fx þ 1; 2x þ 2g and fx þ 2; 2x þ 1g. Regardless of the signs of the edges of the path (u, v, w), if endpoints are colored using colors from two different sets, then we can color the middle point of the path.
Proof of Theorem 3 Let us consider the partition of VðSP À0 9 Þ into four sets: f1; 2g, fx; 2xg, fx þ 1; 2x þ 2g and fx þ 2; 2x þ 1g. Observe that each set contains a pair fa; Àag and that for every vertex v 2 SP À0 9 , both N þ ðvÞ and N À ðvÞ contain an element in at least three sets. We color the grid row by row. Three vertices in the first row we can easily color with SP À0 9 . Assume now that, for k [ 1, the first k À 1 rows of G have been already colored and we color k-th row. Let us denote the 2x + 2 2x + 1 Fig. 2 2-edge-colored graph SP À0 9 . Solid lines represent edges signed by 'þ' vertices in the k À 1-th row by a 1 ; a 2 ; a 3 and the vertices in the k-th row by b 1 ; b 2 ; b 3 . Now color vertices b 1 ; b 2 ; b 3 starting from b 2 . Depending on sign of the edge fa 2 ; b 2 g, for cðb 2 Þ use color in N þ ðcða 2 ÞÞ or N À ðcða 2 ÞÞ which does not belong to the pairs: fcða 1 Þ; Àcða 1 Þg and fcða 3 Þ; Àcða 3 Þg. By Lemma 12, we can extend this coloring to vertex b 1 (using path ða 1 ; b 1 ; b 2 Þ) and to b 3 (using ðb 2 ; b 3 ; a 3 Þ). h

Discussion
In this article, we show that SP 9 colors all 2-edge-colored grids and that SP À0 9 colors all 2-edge-colored grids with 3 columns. Combining this with known lower bounds we have 8 v 2 ðGÞ 9 and 7 v 2 ðG 3 Þ 8. In our opinion, SP À0 9 is the best candidate to lower the upper bound for v 2 ðGÞ. However, we do not know whether SP À0 9 colors all grids, and even whether it colors all grids with 4 columns. On the other hand, to increase lower bound for v 2 ðGÞ we have to prove that there exists a 2edge-colored grid G such that there is no homomorphism G ! SP À0 9 . Finding such a grid also seems to be difficult.
Many authors investigate the relations between 2-edge-coloring and oriented coloring. Oriented coloring (see [9] for a short survey) of an oriented graph G is a homomorphism from G to a tournament H. The oriented chromatic number v o ðGÞ for oriented graph and for undirected graph are define analogously to 2-edgecolored case. A lot of research on 2-edge-colored and oriented coloring used the same techniques and get similar results. For example, in both cases, authors often use Paley graphs (or Paley tournaments) for the target graph. These may result from the similarity of the definitions: in both cases, edges are in one of two states (one of two signs or one of two directions). Despite these similarities, it is known that v o ðGÞ À v 2 ðGÞ and v 2 ðGÞ À v o ðGÞ can be arbitrarily large [3].
For grids, these two colorings seems to behave very similar. For example, for grids with 2 columns, we know that: v 2 ðG 2 Þ ¼ 5 [2] and v o ðG 2 Þ ¼ 6 [5]. For grids with 3 columns, we have 7 v 2 ðG 3 Þ 8 and v o ðG 3 Þ ¼ 7 [10]. The difference between these two chromatic number is very small. Therefore one can hope that also for the class of grids, they are close to each other. We know that oriented chromatic number for grids lies between 8 and 11 [4,5]. Upper bound has not been improved since the first paper on the topic.
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