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Large Monochromatic Components in Two-colored Grids
Electronic Notes in Discrete Mathematics 29 (2007) 3–9 www.elsevier.com/locate/endm
Large Monochromatic Components in Two-colored Grids Jiˇr´ı Matouˇsek 1 Department of Applied Mathematics and Institute of Theoretical Computer Science Charles University Prague, Czech Republic
Aleˇs Pˇr´ıvˇetiv´y 2 Department of Applied Mathematics Charles University Prague, Czech Republic
Abstract Let Dnd denote the d-dimensional grid with diagonals, that is, the graph with vertex set {1, 2, . . . , n}d and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary 2-coloring of the vertices of Dnd there exists a monochromatic connected subgraph with at least n d−1 − d2 nd−2 vertices. We also consider a d-dimensional triangulated grid; this is the graph of a triangulation of the solid cube [1, n]d that refines the subdivision of [1, n]d into the grid of unit cubes. Here every 2-coloring has a monochromatic connected subgraph √ with Ω(nd−1 / d ) vertices. Keywords: graph coloring, HEX lemma, separator
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Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.002
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Introduction
For a graph G and an integer k, we define mcck (G) as the smallest m such that there exists a coloring of the vertices of G by k colors with no monochromatic connected subgraph having more than m vertices. (So the usual chromatic number χ(G) equals min{k : mcck (G) = 1}.) This graph parameter has been investigated in several papers (e.g., [KMR+97], [ADO+03], [HST03], [LMST07] and references therein). Here we consider mcc2 for a geometrically defined class of graphs—cubic grids with diagonals. As a starting point, let us consider a planar graph Tn2 obtained by adding a diagonal to every inner face of an n × n square grid. The well-known planar HEX lemma asserts that for any red-blue coloring of the vertices, there exists a red path connecting the top and bottom sides or a blue path connecting the left and right sides [G79]. In particular, mcc2 (Tn2 ) ≥ n. Linial asked for a d-dimensional analog of this statement: What can be said about mcc2 for a d-dimensional grid of side n? Since the d-dimensional grid in itself is bipartite, we have to add some diagonals in order to get a meaningful question, similar to the planar case. There are at least two reasonable ways of adding diagonals. Triangulated grids. First we consider a d-dimensional triangulated grid Tnd , defined geometrically as follows. We begin with the d-dimensional solid cube [1, n]d and we subdivide it into the grid of unit cubes. Then we triangulate each of the unit cubes in such a way that the simplices of all of these triangulations taken together form a face-to-face triangulation of [1, n]d (i.e., a simplicial complex in the sense of algebraic topology). Then Tnd is the graph of such a triangulation; that is, the vertex set is [n]d (where [n] is the shorthand for the set {1, 2, . . . , n}), and the edges of Tnd are the edges of the triangulation. Thus, for given d and n, Tnd is not defined uniquely, but rather it stands for an arbitrary graph from a (finite) family. The obvious “horizontal layer” coloring, by the parity of the first coordinate, shows that mcc2 (Tnd ) ≤ nd−1 , and one might perhaps suspect this coloring to be optimal. However, for d ≥ 3, there are better colorings at least for some of the possible triangulated grids. d Namely, let us define the ith diagd d d onal layer Ln (i) by Ln (i) = {x ∈ [n] : j=1 xj = i}, and the diagonal-layer coloring by coloring all Ldn (i) with even i red and all Ldn (i) with odd i blue. It can be checked that there are triangulated grids Tnd with no edges connecting Ldn (i) and Ldn (i ) with |i − i | ≥ 2, and for these, the largest monochromatic connected component is the largest diagonal layer. For d = 3, for example, maxi |Ldn (i)| is approximately 43 n2 .
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We prove that the diagonal-layer coloring is not farfrom optimal in the worst case. Let us define, for α ∈ [0, 1], iα = min{i : j≤i |Ldn (j)| > αnd }. (n+1)d It can be checked that i1/2 = and that Ldn (i1/2 ) is either the single 2 largest diagonal layer or one of the two largest diagonal layers. Theorem 1.1 For every d ≥ 3 and n ≥ 3, we have mcc 2 (Tnd ) ≥ |Ldn (i2/3 )| − d−1 n√ |Ld−1 . n (i1/2 )|. The last quantity is of order d We conjecture that the right answer is |Ldn (i1/2 )|; that is, the diagonal-layer coloring is optimal in the worst case. Grid with all diagonals. Next, we define the graph Dnd , the d-dimensional grid with all diagonals, as the graph with vertex set [n]d and edge set {{u, v} : u, v ∈ [n]d , maxi |ui − vi | ≤ 1}. In this case we show that the horizontal-layer coloring is almost optimal (the remaining lower-order term is probably an artifact of the proof method): Theorem 1.2 We have mcc2 (Dnd ) ≥ nd−1 − d2 nd−2 . A different d-dimensional generalization of the planar HEX lemma, appearing, e.g., in Gale [G79], asserts that if the vertices of Tnd are colored by d colors, then there is a monochromatic path connecting two opposite facets of the cube [1, n]d . In particular, mccd (Tnd ) ≥ n. It seems natural to conjecture that mcck (Tnd ) is of order nd−k+1 for d fixed and all k = 3, 4, . . . , d − 1, but this problem appears challenging and the techniques used in the present paper for the case k = 2 do not seem applicable.
2
Outline of the proofs
By a separator in a graph G we mean a subset S ⊆ V (G) whose removal disconnects G into components of size at most 12 |V (G)|. The main part of the proof of both Theorem 1.1 and Theorem 1.2 is the following proposition: Proposition 2.1 Let G = G(K) be the graph of a finite simply connected simplicial complex K. For an arbitrary coloring of V (G) by two colors there exists a monochromatic separator S ⊆ V (G) such that the subgraph induced in G by S is connected. Here by a graph of a simplicial complex K we mean the 1-skeleton (edges of the graph correspond to 1-dimensional simplices of K), and we call K simply connected if the polyhedron of K is simply connected; that is, every closed loop in it can be contracted to a point (see, e.g., [Hat01] or [Mun84] for more background). In particular, the assumptions of Proposition 2.1 obviously
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apply to the graphs Tnd , and we can verify that they also apply to Dnd (here we use an “artificial” 2-dimensional simplicial complex K, which cannot even be embedded in Rd —this part is omitted in this extended abstract). We prove the proposition in the next section. Theorems 1.1 and 1.2 are derived from it using suitable isoperimetric inequalities. For a graph G and a set A ⊆ V (G) the vertex boundary of A in G is the set vert-bdG (A) = {v ∈ V (G) : v ∈ A, {u, v} ∈ E(G) for some u ∈ A}, and the edge boundary of A in G is edge-bdG (A) = {{u, v} ∈ E(G) : u ∈ A, v ∈ A}. A vertex-isoperimetric inequality for G estimates the minimum possible size of vert-bdG (A) as a function of |A|, and similarly for an edge-isoperimetric inequality and edge-bdG (A). d Let Gdn be the graph of the dordinary cube grid; i.e., two vertices u, v ∈ [n] are connected by an edge if i=1 |ui − vi | = 1. Proposition 2.2 For any set A ⊆ [n]d with |vert-bdDnd (A)| ≥ nd−1 − d2 nd−2 .
1 d n 4
≤ |A| ≤
3 d n 4
we have
Proposition 2.3 For every α ∈ (0, 12 ) and every A ⊆ [n]d with αnd ≤ |A| ≤ (1 − α)nd we have |vert-bdGdn (A)| ≥ |Ldn (i1−α )| − |Ld−1 n (i1/2 )|. We derive Proposition 2.2 from an edge-isoperimetric inequality for Gdn due to Bollob´as and Leader [BL91a], stating that |edge-bdGdn (A)| ≥ nd−1 whenever 1 d n ≤ |A| ≤ 34 nd . Proposition 2.3 is derived from a vertex-isoperimetric 4 inequality for Gdn from [BL91b]. The derivations are not too difficult, but rather lengthy and technical, and we omit them in this extended abstract. To prove Theorem 1.1, we consider a red-blue coloring of the vertices of Tnd , and we choose a monochromatic connected separator S in it according to Proposition 2.1. We may assume |S| ≤ 13 nd , for otherwise, we would be done. Then by a standard argument, [n]d \ S can be partitioned into sets A and B so that no edge connects A to B and 31 nd ≤ |A| ≤ 23 nd . Since S ⊇ vert-bdTnd (A) ⊇ vert-bdGdn (A), Proposition 2.3 shows that |S| ≥ |Ldn (i2/3 )| − |Ld−1 n (i1/2 )| as claimed in Theorem 1.1. The asymptotic estimate for this quantity given in the theorem can be obtained from a quantitative version of the central limit theorem due to Feller [Fel43] by a technical calculation. Theorem 1.2 follows from Proposition 2.2 in almost the same way.
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Monochromatic connected separators
Here we prove Proposition 2.1. We will need the following variation of the standard simplicial approximation theorem. Proposition 3.1 Let K and L be simplicial complexes and let f : K → L
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be an arbitrary continuous map of the polyhedron of K in the polyhedron of L. ˜ of K and a simplicial map f˜ of K ˜ into L such Then there is a refinement K that (∗) If σ ∈ K is a simplex such that f (σ) is completely contained in some ˜ lying in σ to vertices of τ . simplex τ ∈ L, then f˜ maps all vertices of K In particular, if f (v) ∈ V (L) for some v ∈ V (K), then f˜(v) = f (v). We recall that a simplicial map of K into L is a map of the vertex sets, V (K) → V (L), such that the image of each simplex of K is a simplex of L. Let us remark that condition (∗) doesn’t appear in the formulations of the simplicial approximation theorem in the literature, but it immediately follows from the standard proof; see, for example, [Hat01] or [Mun84]. Lemma 3.2 Let G be a graph as in Proposition 2.1 with V (G) colored red and blue. Let C be a cycle in G, let u, v ∈ V (C) be two distinct red vertices, and let A and B be the two connected components (arcs) of C \ {u, v}. If there is no blue path from a vertex of A to a vertex of B in G, then u and v are connected by a red path. Proof. Let us choose a triangulation T with T homeomorphic to the unit disk and with the boundary cycle CT such that CT has the same length as C. Let f0 : V (CT ) → V (C) be an isomorphism of the two cycles, and let f : CT → C be the (unique) extension of f0 to a simplicial map. (So here we also regard CT as a 1-dimensional subcomplex of T and C as a 1dimensional subcomplex of K.) Since T is homeomorphic to B 2 and since ¯ T → K. K is simply connected, f can be extended to a continuous map f: Applying our version of the simplicial approximation theorem (Proposition 3.1) to the mapping f¯, we obtain a refinement T˜ of T and a simplicial map f˜ of T˜ into K. Let C˜T be the subcomplex of T˜ that refines CT (so C˜T = CT is the boundary of the disk T ). According to condition (∗) in Proposition 3.1, we have f˜(v) = f (v) for every vertex v of CT , and for every edge (1-simplex) e of CT with vertices u and v, all vertices of C˜T lying in e are mapped by f˜ to either f (u) or f (v). Each vertex of T˜ is mapped to a vertex f˜(v) ∈ V (G) since f˜ is a simplicial map, so we can define the color of each v ∈ V (T˜ ) as the color of ˜ f(v). We note that if x1 , x2 , . . . , xk is a monochromatic path in T˜, then ˜ ˜ ˜ k ) is a monochromatic walk in G. A monochromatic walk f(x1 ), f(x2 ), . . . , f(x can be shortcut to a monochromatic path. Now, by a suitable variant of the HEX lemma mentioned in the introduction (see, e.g., [MN98], Claim 6.1.4 for a very similar statement), since T˜ is not
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connected by a blue path to C˜T in G, there has to exist a red path connecting the two red vertices uT = f −1 (u) and vT = f −1 (v) as required. Proof of Proposition 2.1. If vertices of G are colored with only one color, we can take V (G) for S and we are done. Otherwise, we choose S as the vertex set of a monochromatic connected subgraph of G that is inclusion-maximal (no vertex of the same color can be added), and such that the largest component G1 of G \ S has the smallest possible number of vertices. Without loss of generality, let S be colored blue. We claim that G1 has at most than 21 |V (G)| vertices. For contradiction, we thus assume |V (G1 )| > 12 |V (G)|. We define S as the union of all inclusionmaximal red connected subgraphs of G1 that have at least one vertex connected to S (so S is the “red foam” on the blue surface of S facing G1 ). We will show that S is “better” than S; namely, that S induces an inclusionmaximal connected blue subgraph of G and the largest component of G \ S is smaller than G1 . The set S is red by definition. Next, we show that S induces a connected subgraph (then it will also be clear that it is inclusion-maximal, by the choice of S ). It suffices us to show that any two vertices u and v of S are connected by a red path in G. Moreover, it suffices us to show this only for u and v that are both connected to S by an edge. Let u ∈ S be a blue neighbor of u and let v ∈ S be a blue neighbor of v. Since S is a blue connected subgraph, there is a blue path P1 connecting u and v . Since G1 is connected, u and v are connected by a path P2 in G1 , which is is vertex-disjoint from P1 . The path P2 may consist of vertices of both colors, but there is no blue path from a vertex of P2 to P1 (by the inclusion-maximality of S). The paths P1 and P2 together form a cycle C in G. The graph G and the cycle C satisfy all conditions of Lemma 3.2, and so we have a red path connecting u and v. Hence S indeed induces a connected subgraph. Each component of G \ S is contained either in G1 \ S , or in G \ G1 . We have |V (G1 ) \ S | < |V (G1 )| since S = ∅, and |V (G) \ V (G1 )| < 12 |V (G)|. Thus the largest component of G \ S is strictly smaller than G1 , and this contradicts the choice of S.
References [ADO+03] N. Alon, G. Ding, B. Oporowski, D. Vertigan. Partitioning into graphs with only small components. Journal of Combinatorial Theory (Series B), 87:231–243, 2003. [BL91a] B. Bollob´ as and I. Leader. Compressions and Isoperimetric
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Inequalities. Journal of Combinatorial Theory (Series A) 56:47-62, 1991. [BL91b] B. Bollob´as and I. Leader. Edge-isoperimetric inequalities in the grid. Combinatorica 11(4):299–314, 1991. [Fel43] W. Feller. Generalization of a Probability Limit Theorem of Cramer. Transactions of the American Mathematical Society Vol. 54, No. 3, pp. 361-372, 1943. [G79] D. Gale. The game of hex and the Brouwer fixed-point theorem. Amer. Math. Monthly 86:818–827, 1979. [Hat01] A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge 2001. Electronic version available at http://math.cornell.edu/~hatcher#AT1. [HST03] P. Haxell, T. Szab´ o, G. Tardos. Bounded size components—partitions and transversals. Journal of Combinatorial Theory (Series B) 88:281– 297, 2003. [KMR+97] J. Kleinberg, R. Motwani, P. Raghavan, S. Venkatasubramanian. Storage management for evolving databases. Proc. 38th IEEE Symposium on Foundations of Computer Science, 1997. [LMST07] N. Linial, J. Matouˇsek, O. Sheffet, and G. Tardos. Graph coloring with no large monochromatic components. Submitted, 2007. Extended abstract to appear in Proc. EUROCOMB 2007. [LS93] N. Linial and M. Saks. Low diameter graph decompositions. Combinatorica 13:441–454, 1993. [MN98] J. Matouˇsek and J. Neˇsetˇril. Invitation to Discrete Mathematics. Oxford University Press, Oxford 1998. [Mun84] J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Reading, MA, 1984.
Large Monochromatic Components in Two-colored Grids Jiˇr´ı Matouˇsek 1 Department of Applied Mathematics and Institute of Theoretical Computer Science Charles University Prague, Czech Republic
Aleˇs Pˇr´ıvˇetiv´y 2 Department of Applied Mathematics Charles University Prague, Czech Republic
Abstract Let Dnd denote the d-dimensional grid with diagonals, that is, the graph with vertex set {1, 2, . . . , n}d and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary 2-coloring of the vertices of Dnd there exists a monochromatic connected subgraph with at least n d−1 − d2 nd−2 vertices. We also consider a d-dimensional triangulated grid; this is the graph of a triangulation of the solid cube [1, n]d that refines the subdivision of [1, n]d into the grid of unit cubes. Here every 2-coloring has a monochromatic connected subgraph √ with Ω(nd−1 / d ) vertices. Keywords: graph coloring, HEX lemma, separator
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Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.002
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Introduction
For a graph G and an integer k, we define mcck (G) as the smallest m such that there exists a coloring of the vertices of G by k colors with no monochromatic connected subgraph having more than m vertices. (So the usual chromatic number χ(G) equals min{k : mcck (G) = 1}.) This graph parameter has been investigated in several papers (e.g., [KMR+97], [ADO+03], [HST03], [LMST07] and references therein). Here we consider mcc2 for a geometrically defined class of graphs—cubic grids with diagonals. As a starting point, let us consider a planar graph Tn2 obtained by adding a diagonal to every inner face of an n × n square grid. The well-known planar HEX lemma asserts that for any red-blue coloring of the vertices, there exists a red path connecting the top and bottom sides or a blue path connecting the left and right sides [G79]. In particular, mcc2 (Tn2 ) ≥ n. Linial asked for a d-dimensional analog of this statement: What can be said about mcc2 for a d-dimensional grid of side n? Since the d-dimensional grid in itself is bipartite, we have to add some diagonals in order to get a meaningful question, similar to the planar case. There are at least two reasonable ways of adding diagonals. Triangulated grids. First we consider a d-dimensional triangulated grid Tnd , defined geometrically as follows. We begin with the d-dimensional solid cube [1, n]d and we subdivide it into the grid of unit cubes. Then we triangulate each of the unit cubes in such a way that the simplices of all of these triangulations taken together form a face-to-face triangulation of [1, n]d (i.e., a simplicial complex in the sense of algebraic topology). Then Tnd is the graph of such a triangulation; that is, the vertex set is [n]d (where [n] is the shorthand for the set {1, 2, . . . , n}), and the edges of Tnd are the edges of the triangulation. Thus, for given d and n, Tnd is not defined uniquely, but rather it stands for an arbitrary graph from a (finite) family. The obvious “horizontal layer” coloring, by the parity of the first coordinate, shows that mcc2 (Tnd ) ≤ nd−1 , and one might perhaps suspect this coloring to be optimal. However, for d ≥ 3, there are better colorings at least for some of the possible triangulated grids. d Namely, let us define the ith diagd d d onal layer Ln (i) by Ln (i) = {x ∈ [n] : j=1 xj = i}, and the diagonal-layer coloring by coloring all Ldn (i) with even i red and all Ldn (i) with odd i blue. It can be checked that there are triangulated grids Tnd with no edges connecting Ldn (i) and Ldn (i ) with |i − i | ≥ 2, and for these, the largest monochromatic connected component is the largest diagonal layer. For d = 3, for example, maxi |Ldn (i)| is approximately 43 n2 .
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We prove that the diagonal-layer coloring is not farfrom optimal in the worst case. Let us define, for α ∈ [0, 1], iα = min{i : j≤i |Ldn (j)| > αnd }. (n+1)d It can be checked that i1/2 = and that Ldn (i1/2 ) is either the single 2 largest diagonal layer or one of the two largest diagonal layers. Theorem 1.1 For every d ≥ 3 and n ≥ 3, we have mcc 2 (Tnd ) ≥ |Ldn (i2/3 )| − d−1 n√ |Ld−1 . n (i1/2 )|. The last quantity is of order d We conjecture that the right answer is |Ldn (i1/2 )|; that is, the diagonal-layer coloring is optimal in the worst case. Grid with all diagonals. Next, we define the graph Dnd , the d-dimensional grid with all diagonals, as the graph with vertex set [n]d and edge set {{u, v} : u, v ∈ [n]d , maxi |ui − vi | ≤ 1}. In this case we show that the horizontal-layer coloring is almost optimal (the remaining lower-order term is probably an artifact of the proof method): Theorem 1.2 We have mcc2 (Dnd ) ≥ nd−1 − d2 nd−2 . A different d-dimensional generalization of the planar HEX lemma, appearing, e.g., in Gale [G79], asserts that if the vertices of Tnd are colored by d colors, then there is a monochromatic path connecting two opposite facets of the cube [1, n]d . In particular, mccd (Tnd ) ≥ n. It seems natural to conjecture that mcck (Tnd ) is of order nd−k+1 for d fixed and all k = 3, 4, . . . , d − 1, but this problem appears challenging and the techniques used in the present paper for the case k = 2 do not seem applicable.
2
Outline of the proofs
By a separator in a graph G we mean a subset S ⊆ V (G) whose removal disconnects G into components of size at most 12 |V (G)|. The main part of the proof of both Theorem 1.1 and Theorem 1.2 is the following proposition: Proposition 2.1 Let G = G(K) be the graph of a finite simply connected simplicial complex K. For an arbitrary coloring of V (G) by two colors there exists a monochromatic separator S ⊆ V (G) such that the subgraph induced in G by S is connected. Here by a graph of a simplicial complex K we mean the 1-skeleton (edges of the graph correspond to 1-dimensional simplices of K), and we call K simply connected if the polyhedron of K is simply connected; that is, every closed loop in it can be contracted to a point (see, e.g., [Hat01] or [Mun84] for more background). In particular, the assumptions of Proposition 2.1 obviously
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apply to the graphs Tnd , and we can verify that they also apply to Dnd (here we use an “artificial” 2-dimensional simplicial complex K, which cannot even be embedded in Rd —this part is omitted in this extended abstract). We prove the proposition in the next section. Theorems 1.1 and 1.2 are derived from it using suitable isoperimetric inequalities. For a graph G and a set A ⊆ V (G) the vertex boundary of A in G is the set vert-bdG (A) = {v ∈ V (G) : v ∈ A, {u, v} ∈ E(G) for some u ∈ A}, and the edge boundary of A in G is edge-bdG (A) = {{u, v} ∈ E(G) : u ∈ A, v ∈ A}. A vertex-isoperimetric inequality for G estimates the minimum possible size of vert-bdG (A) as a function of |A|, and similarly for an edge-isoperimetric inequality and edge-bdG (A). d Let Gdn be the graph of the dordinary cube grid; i.e., two vertices u, v ∈ [n] are connected by an edge if i=1 |ui − vi | = 1. Proposition 2.2 For any set A ⊆ [n]d with |vert-bdDnd (A)| ≥ nd−1 − d2 nd−2 .
1 d n 4
≤ |A| ≤
3 d n 4
we have
Proposition 2.3 For every α ∈ (0, 12 ) and every A ⊆ [n]d with αnd ≤ |A| ≤ (1 − α)nd we have |vert-bdGdn (A)| ≥ |Ldn (i1−α )| − |Ld−1 n (i1/2 )|. We derive Proposition 2.2 from an edge-isoperimetric inequality for Gdn due to Bollob´as and Leader [BL91a], stating that |edge-bdGdn (A)| ≥ nd−1 whenever 1 d n ≤ |A| ≤ 34 nd . Proposition 2.3 is derived from a vertex-isoperimetric 4 inequality for Gdn from [BL91b]. The derivations are not too difficult, but rather lengthy and technical, and we omit them in this extended abstract. To prove Theorem 1.1, we consider a red-blue coloring of the vertices of Tnd , and we choose a monochromatic connected separator S in it according to Proposition 2.1. We may assume |S| ≤ 13 nd , for otherwise, we would be done. Then by a standard argument, [n]d \ S can be partitioned into sets A and B so that no edge connects A to B and 31 nd ≤ |A| ≤ 23 nd . Since S ⊇ vert-bdTnd (A) ⊇ vert-bdGdn (A), Proposition 2.3 shows that |S| ≥ |Ldn (i2/3 )| − |Ld−1 n (i1/2 )| as claimed in Theorem 1.1. The asymptotic estimate for this quantity given in the theorem can be obtained from a quantitative version of the central limit theorem due to Feller [Fel43] by a technical calculation. Theorem 1.2 follows from Proposition 2.2 in almost the same way.
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Monochromatic connected separators
Here we prove Proposition 2.1. We will need the following variation of the standard simplicial approximation theorem. Proposition 3.1 Let K and L be simplicial complexes and let f : K → L
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be an arbitrary continuous map of the polyhedron of K in the polyhedron of L. ˜ of K and a simplicial map f˜ of K ˜ into L such Then there is a refinement K that (∗) If σ ∈ K is a simplex such that f (σ) is completely contained in some ˜ lying in σ to vertices of τ . simplex τ ∈ L, then f˜ maps all vertices of K In particular, if f (v) ∈ V (L) for some v ∈ V (K), then f˜(v) = f (v). We recall that a simplicial map of K into L is a map of the vertex sets, V (K) → V (L), such that the image of each simplex of K is a simplex of L. Let us remark that condition (∗) doesn’t appear in the formulations of the simplicial approximation theorem in the literature, but it immediately follows from the standard proof; see, for example, [Hat01] or [Mun84]. Lemma 3.2 Let G be a graph as in Proposition 2.1 with V (G) colored red and blue. Let C be a cycle in G, let u, v ∈ V (C) be two distinct red vertices, and let A and B be the two connected components (arcs) of C \ {u, v}. If there is no blue path from a vertex of A to a vertex of B in G, then u and v are connected by a red path. Proof. Let us choose a triangulation T with T homeomorphic to the unit disk and with the boundary cycle CT such that CT has the same length as C. Let f0 : V (CT ) → V (C) be an isomorphism of the two cycles, and let f : CT → C be the (unique) extension of f0 to a simplicial map. (So here we also regard CT as a 1-dimensional subcomplex of T and C as a 1dimensional subcomplex of K.) Since T is homeomorphic to B 2 and since ¯ T → K. K is simply connected, f can be extended to a continuous map f: Applying our version of the simplicial approximation theorem (Proposition 3.1) to the mapping f¯, we obtain a refinement T˜ of T and a simplicial map f˜ of T˜ into K. Let C˜T be the subcomplex of T˜ that refines CT (so C˜T = CT is the boundary of the disk T ). According to condition (∗) in Proposition 3.1, we have f˜(v) = f (v) for every vertex v of CT , and for every edge (1-simplex) e of CT with vertices u and v, all vertices of C˜T lying in e are mapped by f˜ to either f (u) or f (v). Each vertex of T˜ is mapped to a vertex f˜(v) ∈ V (G) since f˜ is a simplicial map, so we can define the color of each v ∈ V (T˜ ) as the color of ˜ f(v). We note that if x1 , x2 , . . . , xk is a monochromatic path in T˜, then ˜ ˜ ˜ k ) is a monochromatic walk in G. A monochromatic walk f(x1 ), f(x2 ), . . . , f(x can be shortcut to a monochromatic path. Now, by a suitable variant of the HEX lemma mentioned in the introduction (see, e.g., [MN98], Claim 6.1.4 for a very similar statement), since T˜ is not
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connected by a blue path to C˜T in G, there has to exist a red path connecting the two red vertices uT = f −1 (u) and vT = f −1 (v) as required. Proof of Proposition 2.1. If vertices of G are colored with only one color, we can take V (G) for S and we are done. Otherwise, we choose S as the vertex set of a monochromatic connected subgraph of G that is inclusion-maximal (no vertex of the same color can be added), and such that the largest component G1 of G \ S has the smallest possible number of vertices. Without loss of generality, let S be colored blue. We claim that G1 has at most than 21 |V (G)| vertices. For contradiction, we thus assume |V (G1 )| > 12 |V (G)|. We define S as the union of all inclusionmaximal red connected subgraphs of G1 that have at least one vertex connected to S (so S is the “red foam” on the blue surface of S facing G1 ). We will show that S is “better” than S; namely, that S induces an inclusionmaximal connected blue subgraph of G and the largest component of G \ S is smaller than G1 . The set S is red by definition. Next, we show that S induces a connected subgraph (then it will also be clear that it is inclusion-maximal, by the choice of S ). It suffices us to show that any two vertices u and v of S are connected by a red path in G. Moreover, it suffices us to show this only for u and v that are both connected to S by an edge. Let u ∈ S be a blue neighbor of u and let v ∈ S be a blue neighbor of v. Since S is a blue connected subgraph, there is a blue path P1 connecting u and v . Since G1 is connected, u and v are connected by a path P2 in G1 , which is is vertex-disjoint from P1 . The path P2 may consist of vertices of both colors, but there is no blue path from a vertex of P2 to P1 (by the inclusion-maximality of S). The paths P1 and P2 together form a cycle C in G. The graph G and the cycle C satisfy all conditions of Lemma 3.2, and so we have a red path connecting u and v. Hence S indeed induces a connected subgraph. Each component of G \ S is contained either in G1 \ S , or in G \ G1 . We have |V (G1 ) \ S | < |V (G1 )| since S = ∅, and |V (G) \ V (G1 )| < 12 |V (G)|. Thus the largest component of G \ S is strictly smaller than G1 , and this contradicts the choice of S.
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