Online Ramsey Theory for Planar Graphs

An online Ramsey game (G,H) is a game between Builder and Painter, alternating in turns. During each turn, Builder draws an edge, and Painter colors it blue or red. Builder’s goal is to force Painter to create a monochromatic copy of G, while Painter’s goal is to prevent this. The only limitation for Builder is that after each of his moves, the resulting graph has to belong to the class of graphs H. It was conjectured by Grytczuk, Ha luszczak, and Kierstead (2004) that if H is the class of planar graphs, then Builder can force a monochromatic copy of a planar graph G if and only if G is outerplanar. Here we show that the “only if” part does not hold while the “if” part does.


Introduction
For a fixed graph G and a class of graphs H such that G ∈ H, an online Ramsey game (G, H), defined by Grytczuk, Ha luszczak, and Kierstead [5], is a game between Builder and Painter with the following rules.The game starts with the empty graph on infinitely many vertices.On the i-th turn, Builder adds a new edge to the graph created in the first i − 1 turns so that the resulting graph belongs to H (we say that Builder plays on H), and Painter colors this edge blue or red.Builder wins if he can always force Painter to create a monochromatic copy of G (or force G for short).We then say that G is unavoidable on H.A graph G is unavoidable if it is unavoidable on planar graphs.On the other hand, if Painter can ensure that a monochromatic copy of G is never created, then G is avoidable on H.A class of graphs H is self-unavoidable if every graph of H is unavoidable on H.
According to Ramsey's theorem, for every t ∈ N there exists n ∈ N such that every 2-coloring of the edges of K n contains a monochromatic copy of K t .Thus, without the electronic journal of combinatorics 21(1) (2014), #P1.64 restricting to H, Builder would always win the online Ramsey game by creating sufficiently large complete graph.The size Ramsey number r(G) for a graph G is the minimum number of edges of a graph that contains a monochromatic copy of G in every 2-coloring of its edges.The online size Ramsey number r(G) is the minimum m such that Builder can force G by playing on the class of graphs with at most m edges.Clearly, r(G) r(G) (Builder wins by presenting a graph of size r(G) that contains a monochromatic copy of G for any 2-edge-coloring).However, Builder may be able to win using less than r(G) edges since he can adapt his strategy to Painter's coloring.One can then ask whether or not r(G) = o(r(G)).The basic conjecture in the field, attributed to Rödl by Kurek and Ruciński [9], is that r(K t ) = o(r(K t )).In 2009, Conlon [3] showed that r(K t ) 1.001 −t (r(K t )) for infinitely many t.On the other hand, if G is a path or a cycle, then both r(G) and r(G) are linear in |V (G)| (see [1], [6], [7]).
Butterfield et al. [2] studied Online Ramsey games played on the class S k of graphs with maximum degree at most k.The authors introduce an online degree Ramsey number r (G) as the least k for which G is unavoidable on S k .
Online Ramsey games played on various classes of graphs were studied by Grytczuk et al. [5].They proved that the class of k-colorable graphs as well as the class of forests are self-unavoidable.(It was later shown by Kierstead and Konjevod [8] that the k-colorable graphs are self-unavoidable even if Painter uses more colors.)Various games played on planar graphs were investigated in [5].It was shown, for example, that every cycle, as well as the graph K 4 − e, is unavoidable on planar graphs.They made the following conjecture: Conjecture ( [5]).The class of graphs unavoidable on planar graphs is exactly the class of outerplanar graphs.
Here we show that the conjecture is only partially true.In particular, it is true that the class of outerplanar graphs is a subclass of the class of graphs unavoidable on planar graphs.
Theorem 1.Every outerplanar graph is unavoidable on planar graphs.
However, we show that there exists an infinite family of planar but not outerplanar graphs which are unavoidable on planar graphs.Let θ i,j,k denote the union of three internally disjoint paths of lengths i, j, k, respectively.Theorem 2. The graph θ 2,j,k is unavoidable for even j, k.
The paper is organized as follows.In Section 2, we introduce notation.Section 3 gives a proof of Theorem 1, and Section 4 gives a proof of Theorem 2.

Notation
In this section, we first mention several notions that are particularly important for the next discussion.Besides these, we follow standard graph theory terminology (see Diestel [4]).
G : All graphs considered here are simple and undirected.For a graph G, the set of vertices is denoted by V (G) and the set of edges by E(G).The length of a path is the number of its edges.If we replace an edge e of G with a path of length k + 1 (i.e.we place k vertices of degree 2 on e), then we say that e is subdivided k-times.For a fixed graph G, a copy When we say that a graph is a disjoint union of G 1 and G 2 , we are automatically assuming that V 1 ∩ V 2 = ∅.A planar graph is a graph that can be drawn in the plane without edge crossings.An outerplanar graph is a planar graph that can be embedded so that all its vertices belong to the boundary of the outer face.A red-blue graph is a graph with its edges colored red or blue.A red-blue graph will often use the same name as its underlying (uncolored) graph.
Let G 1 and G 2 be two disjoint graphs containing cliques G 2 is a graph formed from the disjoint union of G 1 and G 2 by identifying the vertex v j (G 1 ) with v j (G 2 ) for each j = 1, . . ., k.To simply notation, we write G 1 ⊕G 2 if k 2. Note that G 1 ⊕ G 2 does not specify the appending cliques, and so it is not a well-defined operation.However, if k = 1, then we can make this notation precise and specify the appending vertex v by writing G 1 ⊕ v G 2 (which we will do often).For k = 2, we sometimes write G 1 ⊕ e G 2 , where e(G i ) is a non-oriented edge v 1 (G i )v 2 (G i ) (the resulting graph is again not always unique).Also, we abbreviate (( Let G be a graph, H a subgraph of G.If there exist planar graphs X 1 , . . ., X n such that G = H ⊕ X 1 ⊕ • • • ⊕ X n , then we say that G is reducible to H, and we write G H. It is a well known fact that for k 2, a k-sum of two planar graphs is planar, thus the following holds: Remark 3. If H is a planar graph, and a graph G is reducible to H, then G is planar.
Informally, G is reducible to H if G can be formed from H by successively "appending" planar graphs on edges/vertices.So, Remark 3 says that if the starting graph H is planar, then so is G.
Consider an online Ramsey game on planar graph.A strategy (for Builder) X is a finite sequence of rules that tell Builder how to move on any given turn of the game, no matter how Painter plays.If a monochromatic copy of the target graph G arises, the game ends and Builder wins (provided that the final red-blue graph is planar).The output graph of strategy X is then the final red-blue graph with a fixed monochromatic copy of G, called a winning copy (of G by X) and denoted simply by G if no confusion can arise.This winning copy adopts all notation from the target graph.For example, for a target graph G with vertices u, v and a cycle C, the two corresponding vertices and the corresponding cycle are again denoted by u, v, C in the chosen winning copy G.If Builder always wins when following strategy X, then we say that G is unavoidable by strategy X.The set of all output graphs of a strategy X is denoted X (the calligraphic version of the name of the strategy).

Outerplanar graphs
In this section we show that every outerplanar graph is unavoidable on the class of planar graphs.The idea behind our proof is based on the inductive proof of the self-unavoidability of forests presented by Grytczuk et al. [5].Suppose that Builder's goal is to force a forest T .We can assume that T is a tree (since every forest is contained in some tree).Choose a leaf u of T , let v be the neighbor of u in T , and let T = T − u.Builder forces 2|T | − 1 monochromatic copies of T (where the corresponding final graphs are pairwise disjoint), from which at least |T | are of the same color, say blue.On those copies, Builder builds a new copy of T by adding edges between copies of v.If any one of the new added edges is blue, then that edge and a blue copy of T appended to one of its endpoints form a blue copy of T .Otherwise, those edges form a red copy of T .We will call this strategy the tree strategy.
Since trees are planar, the tree strategy shows that forests are unavoidable (on planar graphs).Moreover, a generalized version of the tree strategy can be used for forcing a graph formed from a tree T by appending a copy of an unavoidable graph G to each vertex of T .Before presenting this strategy we need some notation.
Let T be a tree on vertices v 1 (T ), . . ., v n (T ), and let G be a graph with an arbitrary vertex labeled by v.The ordered triple (T, G, v) denotes the graph Next, let S be any set of red-blue graphs X such that each has a fixed monochromatic copy G X of G. Let A be a red-blue graph with a fixed monochromatic subgraph (T, In our proofs we take S to be the set of all final graphs of some strategy.For example, the set of all output graphs X of a strategy X is a set of red-blue graphs, each with a fixed monochromatic copy of G, and so, for any given tree T , we can talk about (T, X )-reducible graphs.
Suppose that G is unavoidable by strategy X.We consider the following Builder's strategy for forcing a monochromatic copy of (T, G, v).
Figure 2: Forcing a monochromatic copy of (T, G, v), where T is a path of length 2, G a triangle, and v is an arbitrary vertex of V (G).
strategy A (T, G, v, X) of the same color, and in ith of them label the vertex that corresponds to the neighbor of u in T by u i .Add an edge e ij = u i u j if and only if v i v j is an edge in (T, G, v).
To prove that (T, G, v) is unavoidable by strategy A(T, G, v, X), we have to ensure that no matter how Painter plays, a monochromatic copy of the target graph (T, G, v) eventually appears, and that the final graph is planar.Both parts are shown below using induction and reduction arguments that rely on Remark 3. Lemma 4. Let T be a tree, G a graph, and v a vertex of V (G).If G is unavoidable by strategy X, then (T, G, v) is unavoidable by strategy A(T, G, v, X), and every graph Proof.We use all the notation introduced in strategy A. The proof is by induction on the number n of vertices of T .If n = 1, then (T, G, v) = G, which is unavoidable by strategy X by the assumption.Since A(T, G, v, X) = X , the graph A is (T, X )-reducible.Now let n > 1.The following two cases can arise.
Case 1: All edges e ij are red.These edges form a red (T, G, v).Every final graph for forcing H i is planar by the induction hypothesis.Observe that each such graph is appended to (T, G, v) by one vertex only.Thus, A is reducible directly to (T, G, v), and hence is (T, X )-reducible, which proves the planarity as well as the second part of the claim.See Figure 2.
Case 2: Some edge e ij is blue.The graph H i , the edge e ij , and one copy of G contained in H j form a blue (T, G, v).The graph A is planar by previous discussion, so the first part of the claim is complete.Let A i , A j ∈ A(T , G, v, X) be subgraphs of A that were used for forcing H i and H j , respectively.By the induction hypothesis, A i is (T , X )-reducible.
the electronic journal of combinatorics 21(1) (2014), #P1.64 Similarly A j is (T , X )-reducible, and therefore is (u , X )-reducible.Since the rest of A is reducible to e ij , and e ij shares with each of H i and H j only one vertex (the vertex u i and u j , respectively), we get the second part of the claim.
A block is a maximal 2-connected subgraph.For a graph G with a vertex set V = {v 1 , . . ., v k } and blocks B 1 , . . ., B l , the complete block graph B(G) is a graph on V ∪ {B 1 , . . ., B l } formed by the edges v i B j with v i ∈ V (B j ) (see Figure 3).Notice that B(G) can be obtained from the block graph B(G) of G by adding edges with one endpoint of degree 1, and thus, B(G) is a tree for every connected graph G.
Remark 5.The union of an outerplanar graph G and its complete block graph B(G) is planar.
Let H be an outerplanar graph.The weak dual H * of H is the graph obtained from the plane dual of H by removing the vertex that corresponds to the outer face of H.It is easy to see that H * is a forest, which is a tree whenever H is 2-connected.If there exists a vertex r ∈ V (H * ) such that H * rooted in r (denoted by H * (r)) is a full binary tree, then we call H a full outerplanar graph.The height h(H) of a full outerplanar graph H is the number of levels in its full binary tree H * (r).The edge of a full outerplanar graph H incident to the face that corresponds to r, as well as to the the outer face, is the central edge e H of H (see Figure 4, left).For the sake of convenience, a graph that consists of a single edge is also considered to be full outerplanar.Its height is then defined to be 0 and its central edge is the only edge of the graph.Lemma 6.For every outerplanar graph G there exists a full outerplanar graph H such that G ⊆ H.
Proof.Let G T be an almost triangulation of G, i.e. an outerplanar graph formed by triangulating the inner faces of G.The maximum degree of G * T is at most 3, and there exists a vertex r ∈ V (G * T ) of degree 1 or 2. Let H * (r) be a full binary tree of height h(G * T (r)) containing G * T .The graph H is then the desired full outerplanar graph.Recall that for a tree T with n vertices and m edges, a graph G, and a vertex v ∈ V (G), we have (T, Let H be a full outerplanar graph, and for i = 1, . . ., m, let H i be a copy of H with the central edge e i (H i ).Then we define (T, G, v, H) as a graph , (T, G, v, H) is simply the graph that arises from (T, G, v) if we "glue" a copy of H by its central edge to every edge of T (cf. Figure 4, right).
We now present a strategy B for forcing a monochromatic copy of (T, G, v, H), assuming that G is unavoidable by a strategy X. strategy B (T, G, v, H, X)
3. Choose a leaf u of T and call its neighbor u .Call strategy B(T , G , v , H , X ), where • H is the full outerplanar graph of height h − 1, Let {u 1 , . . ., u k } be the vertex set of (T, G, v, H), and thus also a subset of a vertex set of B(T, G, v, H) = T .Adopt this notation to the subgraph T of the winning copy (T , G , v , H ) found by strategy B(T , G , v , H , X ).Add an edge e ij = u i u j in (T , G , v , H ) if and only if u i u j is an edge in (T, G, v, H).
Let S be a set of red-blue graphs such that each X ∈ S contains a fixed monochromatic graph G. Then we set S = S ∪ {G ∪ B(G)}, where G is the fixed monochromatic graph.
Claim.Let T be a tree, G an outerplanar graph, and v ∈ V (G).If G is unavoidable by strategy X, then (T, G, v, H) is unavoidable by strategy B(T, G, v, H, X), and every graph B ∈ B(T, G, v, H, X) is (T, X )-reducible.
This statement implies Theorem 1 since every outerplanar graph G is contained in some full outerplanar graph H by Lemma 6, which can be written as (e H , ({v}, ∅), v, H), and is therefore unavoidable by the above claim.
We adopt all the notation used in strategy B. Let S be the set of all 2-tuples (h, t) ∈ (N ∪ {0}) × N. On S, we define the lexicographic order , i.e. (h 1 , t 1 ) (h 2 , t 2 ) exactly when h 1 < h 2 , or h 1 = h 2 and t 1 t 2 for all h 1 , h 2 ∈ N ∪ {0} and t 1 , t 2 ∈ N. The set S together with the relation is linear, and we can apply induction.
We start with the basis.Suppose first that h 0, and t = 1.Then (T, G, v, H) = G and the claim is trivially satisfied.Let now h = 0, and t 1.In this case we have (T, G, v, H) = (T, G, v).By Lemma 4, (T, G, v) is unavoidable by A(T, G, v, X).So, every graph of A(T, G, v, X) is (T, X )-reducible, and thus (T, X )-reducible since X ⊆ X .
Suppose now that h 1, t 2. By the induction hypothesis ((h, t − 1) (h, t)), G = (T − u, G, v, H) is unavoidable by strategy X = B(T − u, G, v, H, X), and every graph of X is (T − u, X )-reducible.Since G is unavoidable by strategy X , it holds by the induction hypothesis ((h − 1, t) (h, t)) that (T , G , v , H ) is unavoidable by strategy B = B(T , G , v , H , X ), and every graph B of B is (T , X )-reducible.Say that the winning copy (T , G , v , H ) in B is blue.We distinguish the following two cases: Case 1: All edges e ij are red.These edges form a red copy of (T, G, v, H).The graph B is (T , X )-reducible, and thus reducible to T = B(T, G, v, H).Since B arose from B by adding the edges forming (T, G, v, H), B is reducible to (T, G ∪ B(G), v), and thus (T, X )-reducible.
the electronic journal of combinatorics 21(1) (2014), #P1.64 Case 2: At least one edge e ij = u i u j is blue.The endpoints of e ij are connected by a path P in T of length 2. There is a copy of H appended along each of the edges of P .Those two copies of H together with e ij form a full outerplanar graph H of height h with central edge e ij (see Figure 5).Let G i and G j be the blue copies of G = (T − u, G, v, H) appended to u i and u j , respectively.Then H, G i , and the copy of G in G j that is appended to u j form a blue copy of (T, G, v, H).We can assume that Builder chooses this copy as the winning copy.We now prove the second part of the claim.Recall that B is (T , X )-reducible.Let X i , X j be the graphs of X appended to u i , u j , respectively.So, X i is (T − u, X )-reducible, and X j is ({u j }, X )-reducible.Since the rest of the graph B is reducible to e ij , we find that B is (T, X )-reducible.See the diagram on the right in Figure 5.

Non-outerplanar graphs
We now show that an infinite subclass of theta-graphs is unavoidable on planar graphs.Recall that a theta-graph (θ-graph) is the union of three internally disjoint paths that have the same two end vertices.We write θ i,j,k for the theta-graph with paths of length i, j, k.For example, K 2,3 is the graph θ 2,2,2 .
Before stating the main theorem, we introduce a strategy for forcing even cycles.The unavoidability of cycles was proven in [5], but here we need the final graph to have a special type of plane embedding that we utilize in the proof of the main theorem.
Let C be a cycle of even length n that is unavoidable by strategy X.If for every graph X of X there is a plane embedding of X such that (G1) all vertices of V (C) belong to the boundary of one common face, and (G2) there exists a path P ⊂ C of length n 2 such that all vertices of V (P ) lie the boundary of another face, then we say that strategy X is a good strategy.The path P is then called a good path in C.
2. In P , Connect the vertices v 0 and v a 2 by an edge e.

If e has the other color than P , add the path
Lemma 7. Let C be an even cycle.Then strategy C(C) is a good strategy.Proof.We will follow the notation introduced in strategy C. We fix a planar embedding of an output graph of strategy C(C) as shown in Figure 6.By [5], every final graph of the tree strategy for forcing P is a forest, which is reducible to (the chosen monochromatic) P .Assume that P is blue.The following two cases can arise.

Case 1:
The edge e is red.If Painter colors some edge of P blue, a blue copy of C arises since there is a blue path of length n − 1 between such two vertices.Otherwise, P ∪ e is a red cycle C of length n.In both cases, all vertices of the monochromatic copy of C belong to two common faces.See Figure 6, left.
Case 2: The edge e is blue.Suppose that Painter colors some edge e of C − v b v (a 2 +b) blue.Since each such pair is connected by a blue path of length a = n − 1, a blue cycle of length n arises.Condition (G1) is then satisfied by the face bounded by this cycle, and (G2) is satisfied by the face f bounded by the cycle v 0 v a v 2a . . .v a 2 v 0 , which contains a good path on n/2 + 1 vertices if e = v a(a−1) v a 2 and on all n vertices in all the other cases.Suppose now that Painter colors the edge v b v (a 2 +b) blue.Then the blue copy of C is formed by this edge and the blue path starting at v b , going through e, and ending at v (a 2 +b) .All of the vertices of the blue copy of C belong to the outer face, and there is a good path v b v b−1 . . .v 0 v a 2 of length b + 1 = n 2 that belongs to f .Consider the last possibility when C is red.Now, all of the vertices of V (C) belong to the boundary of f , and all but the vertex v (a 2 +b) of V (C) belong to the boundary of f .See Figure 6, right.
Theorem 2. The graph θ 2,j,k is unavoidable for even j, k.
Proof.For fixed j and k, let j = j 2 , k = k 2 .We consider disjoint cycles C 1 , . . ., C j +k +1 of length k + 2. In ith of them, we label an arbitrary vertex by c i and one of the two vertices in distance 2 from c i by v 0 (C i ) if i j + 1, and by v 1 (C i ) otherwise.Let P 1 , . . ., P j +k +2 be paths of length j − 1, where in each P i , one end is labeled by p i , and another one by v 0 (P i ) if i j + 1, and by v 1 (P i ) otherwise.Let Then we write H for a graph that is formed from the disjoint union of L and R by identifying p 1 with p j +k +2 , and p j +1 with p j +2 (see Figure 7, left).The cycle consisting of the paths P 1 , P j +1 , P j +2 , and P j +k +2 is denoted C 0 .
the electronic journal of combinatorics 21(1) (2014), #P1.64 H: θ 2,2,4 : Observe first that having a monochromatic copy of a H, Builder could easily force θ 2,j,k (cf. Figure 7).The graph H is outerplanar, and hence unavoidable by Theorem 1.The problem is that by connecting the proper vertices of the monochromatic copy of H in the resulting graph, the planarity condition would be violated.Therefore, we have to change the strategy for forcing H.
For n = 0, . . ., j So, the graph G j +k +1 is the graph H without the paths P 2 , . . ., P j , P j +3 , . . ., P j +k +1 .Let us refer to the blocks of G n and the corresponding vertices of the complete block graph simply by C 0 , C 1 , . . ., C n .Next, let V be the set of vertices of G n (and thus also of B(G n )) for which the distance from v 0 in G n is even.For G n , we define a subdivided complete block graph B S (G n ) as a graph that arises from B(G n ) by subdividing each edge joining C i (i = 1, . . ., n) and a vertex of V (k − 1)-times.Observe that B S (G) is a tree, and that G ∪ B S (G) is planar.
We now present strategy D for forcing G n .
strategy D (G n ) 1.If n = 0, call strategy C(C 0 ).In C 0 , find a good path P 0 , denote the middle vertex of P 0 by v 0 and its opposite vertex in C 0 by v 1 .We show by induction on n that G n is unavoidable by strategy D(G n ), and that every graph D of D(G n ) can be embedded in the plane so that (1) all vertices v 0 , v 1 , p 1 , p j +1 , c 1 , . . ., c n belong to some face f 1 , and

If
(2) (a) the vertices v 0 , p 1 , p j +1 belong to some face f 2 , other than f 1 , or (b) there is a path P = p 1 c 1 p j +1 of the other color than G n .
The base case is n = 0.By Lemma 7, strategy C(C 0 ) is a good strategy, i.e. every graph of C(C 0 ) can be embedded in such a way that all vertices of C 0 belong to one common face, and there is a path P 0 ⊂ C 0 of length 4(j−1) face and either v 0 , p 1 , p j +1 belong to another common face or there is red path of p 1 c 1 p j +1 .Also, D can be embedded so that all vertices of C n lie in the boundary of a common face.Now, G can be drawn inside that face, which gives both Condition (1) and Condition (2).
2. Add edges of the cycle p 1 c 2 p 2 c 3 . . .p j +1 (= p j +2 )c j +2 p j +3 . . .c j +k +1 p 1 to H.If there is not the path p 1 c 1 p j +1 , also add the edges p 1 c 1 and c 1 p j +1 .
As a consequence of Lemma 4, every graph of A(T 1 , (T 0 , G, v 0 ), v 1 , X) can be embedded in such a way that Conditions (1) and ( 2) hold for G, and that all the vertices p 2 , . . ., p j +k +1 lie in the boundary of the face f 1 .This means that adding the cycle in Step 3 of strategy E does not violate the planarity of the final graph.Finally, Condition (2) ensures that either there already is a path of length 2 connecting p 1 and p j +1 of the desired color, or Builder can add it by connecting p 1 to c 1 and c 1 to p j +1 .

Further problems
The question of whether the class of planar graphs is self-unavoidable is still open.To disprove it, it suffices to find a single planar graph G such that Painter can ensure that a monochromatic copy of G never occurs when playing on planar graphs.The graph K 4 seems to be a good candidate.Conjecture 8. K 4 is avoidable on the class of planar graphs.
Unfortunately, Painter's winning strategies seem to be much harder to find.So far, only one such strategy has been presented; namely a strategy showing that a triangle is avoidable on the class of outerplanar graphs given in [5].

Figure 3 :
Figure 3: A graph G and its complete block graph B(G).

Figure 4 :
Figure 4: A full outerplanar graph H with its full binary tree H * (r) of height 3 (left).The structure of (T, G, v, h) (right).

Figure 5 :
Figure 5: Forcing a monochromatic copy of (T, G, v, H), where T is a path of length 2, G is a cycle of length 4, v is any vertex of V (G), and H is the full outerplanar graph of height 1.

2 Figure 6 :
Figure 6: Forcing cycle of length 4 by strategy C.

3 .
In (T , G , v ), connect two vertices of T = B S (G n ) by an edge if and only if the corresponding vertices are connected by an edge in G n .

Figure 8 :
Figure 8: Left: The graphs G 3 (dashed) and B S (G 3 ) (solid) for j = 2, k = 2. Black vertices represent the vertices of B(G 3 ) whereas white ones are the subdividing vertices.Right: Forcing G 3 , Case 2 -one of the edges added to (T , G , v ) = (B S (G 3 ), G 2 , v 2 ) is blue.