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A note on Gallai–Ramsey number of even wheels
Discrete Mathematics 343 (2020) 111725
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A note on Gallai–Ramsey number of even wheels ∗
Zi-Xia Song a ,1 , Bing Wei b , Fangfang Zhang c,a , ,2 , Qinghong Zhao b a
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Department of Mathematics, University of Mississippi, University, MS 38677, USA c Department of Mathematics, Nanjing University, Nanjing 210093, PR China b
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info
Article history: Received 28 January 2019 Received in revised form 30 October 2019 Accepted 31 October 2019 Available online xxxx Keywords: Gallai coloring Gallai-Ramsey number Rainbow triangle
a b s t r a c t A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses k colors. Given a graph H and an integer k ≥ 1, the Gallai–Ramsey number GRk (H) of H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. Let Wn denote a wheel on n + 1 vertices. In this note, we study Gallai–Ramsey number of W2n and completely determine the exact value of GRk (W4 ) for all k ≥ 2. © 2019 Elsevier B.V. All rights reserved.
1. Introduction All graphs in this paper are finite, simple and undirected. Given a graph G and a set S ⊆ V (G), we use |G| to denote the number of vertices of G, and G[S ] to denote the subgraph of G obtained from G by deleting all vertices in V (G) \ S. For two disjoint sets A, B ⊆ V (G), A is complete to B in G if every vertex in A is adjacent to all vertices in B. We use Kn , Cn , Pn to denote the complete graph, cycle, and path on n vertices, respectively; and Wn to denote a wheel on n + 1 vertices. For any positive integer k, we write [k] for the set {1, . . . , k}. We use the convention ‘‘S :=’’ to mean that S is defined to be the right-hand side of the relation. Given an integer k ≥ 1 and graphs H1 , . . . , Hk , the classical Ramsey number R(H1 , . . . , Hk ) is the least integer N such that every k-coloring of the edges of KN contains a monochromatic copy of Hi in color i for some i ∈ [k]. When H = H1 = · · · = Hk , we simply write Rk (H). Ramsey numbers are notoriously difficult to compute in general. Very little is known for Ramsey numbers of wheels and the exact value of R2 (Wn ) is only known for n ∈ {3, 4, 5} (see [7,11,14]). In this paper, we study Ramsey number of wheels in Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles (that is, a triangle with all its edges colored differently). Gallai colorings naturally arise in several areas including: information theory [15]; the study of partially ordered sets, as in Gallai’s original paper [10] (his result was restated in [13] in the terminology of graphs); and the study of perfect graphs [4]. More information on this topic can be found in [8,9]. A Gallai k-coloring is a Gallai coloring that uses at most k colors. We use (G, τ ) to denote a Gallai k-colored complete graph if G is a complete graph and τ : E(G) → [k] is a Gallai k-coloring. Given an integer k ≥ 1 and graphs H1 , . . . , Hk , the Gallai-Ramsey number GR(H1 , . . . , Hk ) is the least integer N such that every (KN , τ ) contains a monochromatic copy of Hi in color i for some i ∈ [k]. When H = H1 = · · · = Hk , we simply write GRk (H). Clearly, GRk (H) ≤ Rk (H) for all k ≥ 1 and GR(H1 , H2 ) = R(H1 , H2 ). In 2010, Gyárfás, Sárközy, Sebő and Selkow [12] proved the general behavior of GRk (H). ∗ Corresponding author at: Department of Mathematics, Nanjing University, Nanjing 210093, PR China. E-mail addresses: [email protected] (Z.-X. Song), [email protected] (F. Zhang). 1 Supported by the National Science Foundation, USA under Grant No. DMS-1854903. 2 The work is done while the third author was studying at the University of Central Florida as a visiting student, supported by the Chinese Scholarship Council, PR China. https://doi.org/10.1016/j.disc.2019.111725 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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Z.-X. Song, B. Wei, F. Zhang et al. / Discrete Mathematics 343 (2020) 111725
Theorem 1.1 ([12]). Let H be a fixed graph with no isolated vertices and let k ≥ 1 be an integer. Then GRk (H) is exponential in k if H is not bipartite, linear in k if H is bipartite but not a star, and constant (does not depend on k) when H is a star. It turns out that for some graphs H (e.g., when H = C3 ), GRk (H) behaves nicely, while the order of magnitude of Rk (H) seems hopelessly difficult to determine. It is worth noting that finding exact values of GRk (H) is far from trivial, even when |H | is small. The following structural result of Gallai [10] is crucial in determining the exact value of Gallai-Ramsey number of a graph H. Theorem 1.2 ([10]). For any Gallai coloring τ of a complete graph G with |G| ≥ 2, V (G) can be partitioned into nonempty sets V1 , V2 , . . . , Vp with p > 1 so that at most two colors are used on the edges in E(G) \ (E(V1 ) ∪ · · · ∪ E(Vp )) and only one color is used on the edges between any fixed pair (Vi , Vj ) under τ , where E(Vi ) denotes the set of edges in G[Vi ] for all i ∈ [p]. The partition given in Theorem 1.2 is a Gallai partition of the complete graph G under τ . Given a Gallai partition V1 , . . . , Vp of the complete graph G under τ , let vi ∈ Vi for all i ∈ [p] and let R := G[{v1 , . . . , vp }]. Then R is the reduced graph of G corresponding to the given Gallai partition under τ . By Theorem 1.2, all edges in R are colored by at most two colors under τ . One can see that any monochromatic copy of H in R will result in a monochromatic copy of H in G under τ . It is not surprising that Gallai-Ramsey number GRk (H) is closely related to the classical Ramsey number R2 (H). The following is a conjecture of Fox, Grinshpun and Pach [8] on GRk (H) when H is a complete graph. Conjecture 1.3 ([8]). For all k ≥ 1 and t ≥ 3, (R2 (Kt ) − 1)k/2 + 1
{ GRk (Kt ) =
(t − 1)(R2 (Kt ) − 1)
if k is even
(k−1)/2
+ 1 if k is odd.
The first case t = 3 of Conjecture 1.3 follows directly from a result of Chung and Graham [6] from 1983, and a simpler proof can be found in [12]. Recently, the case t = 4 of Conjecture 1.3 was proved in [18]. In this note, we study Gallai–Ramsey number of even wheels. Recall that (G, τ ) denotes a Gallai k-colored complete graph if G is a complete graph and τ : E(G) → [k] is a Gallai k-coloring. For any A, B ⊆ V (G), if all the edges between A and B are colored the same color, say blue, under τ , we say that A is blue-complete to B in (G, τ ). We simply say a is blue-complete to B in (G, τ ) when A = {a}. We first prove the following general lower bound for GRk (W2n ) for all k ≥ 2 and n ≥ 2. Proposition 1.4.
For all n ≥ 2 and k ≥ 2, (R2 (W2n ) − 1) · 5(k−2)/2 + 1
{ GRk (W2n ) ≥
if k is even
2(R2 (W2n ) − 1) · 5(k−3)/2 + 1 if k is odd.
Proof. Let n ≥ 2 be given. For all k ≥ 2, let
{ f (k) :=
(R2 (W2n ) − 1) · 5(k−2)/2
if k is even
2(R2 (W2n ) − 1) · 5(k−3)/2
if k is odd.
For all k ≥ 2, we next construct (Gk , τk ) recursively as follows, where Gk is a complete graph on f (k) vertices and τk : E(Gk ) → [k] is a Gallai k-coloring such that (Gk , τk ) has no monochromatic copy of W2n . For k = 2, let m := f (2) = R2 (W2n ) − 1 and G2 := Km . By the definition of R2 (W2n ), there exists τ2 : E(G2 ) → {1, 2} such that G2 has no monochromatic copy of W2n under τ2 . Then (G2 , τ2 ) is a Gallai 2-colored complete graph on f (2) vertices with no monochromatic copy of W2n , as desired. When k ≥ 3 is even, let (Gk−2 , τk−2 ) be the construction on f (k − 2) vertices with colors in [k − 2] and let (H , c) := (K5 , c) be such that edges of H are colored by colors k and k − 1 with no monochromatic copy of K3 . Let (Gk , τk ) be obtained from (H , c) by replacing each vertex of H with a copy of (Gk−2 , τk−2 ) such that for every uv ∈ E(H), all the edges between the corresponding copies of (Gk−2 , τk−2 ) for u and v are colored by the color c(uv ). Since (Gk−2 , τk−2 ) is a Gallai (k − 2)-colored complete graph with no monochromatic copy of W2n , we see that τk is indeed a Gallai k-coloring and (Gk , τk ) has no monochromatic copy of W2n . When k ≥ 3 is odd, let (Gk−1 , τk−1 ) be the construction on f (k − 1) vertices with colors in [k − 1] and let (Gk , τk ) be the join of two disjoint copies of (Gk−1 , τk−1 ), with all the new edges colored by the color k. In both cases, (Gk , τk ) has no rainbow triangle and no monochromatic copy of W2n . Hence, GRk (W2n ) ≥ f (k) + 1, as desired. ■ We will make use of the following result of Hendry [14] on the exact value of R2 (W4 ). Theorem 1.5 ([14]). R2 (W4 ) = 15. The main result in this note is Theorem 1.6, which establishes the exact value of GRk (W4 ) for all k ≥ 2. Theorem 1.6. For all k ≥ 2,
{ GRk (W4 ) =
14 · 5
k−2 2
28 · 5
k−3 2
+ 1, + 1,
if k is even if k is odd.
Z.-X. Song, B. Wei, F. Zhang et al. / Discrete Mathematics 343 (2020) 111725
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With the support of Theorem 1.6, it seems that the lower bound given in Proposition 1.4 is also the desired upper bound for GRk (W2n ). We propose the following conjecture. Conjecture 1.7. For all k ≥ 2 and n ≥ 2,
{ GRk (W2n ) =
(R2 (W2n ) − 1) · 5
k−2 2
2(R2 (W2n ) − 1) · 5
k−3 2
+ 1, + 1,
if k is even if k is odd.
It is worth noting that very recently, Gallai-Ramsey number of some other graphs with at most five vertices were studied by Li and Wang [17]. More recent work on Gallai-Ramsey numbers of cycles can be found in [1–3,5,16,19,20]. 2. Proof of Theorem 1.6 Let f (1) := 4 and for each integer s ≥ 2, let
{ f (s) :=
14 · 5
s−2 2
28 · 5
s−3 2
, ,
if s is even if s is odd.
Then for all s ≥ 3,
{ f (s) ≥
2f (s − 1) 5f (s − 2).
(⋆)
By Proposition 1.4, it suffices to show that for all k ≥ 2, GRk (W4 ) ≤ f (k) + 1. By Theorem 1.5, R2 (W4 ) = 15 and so GR2 (W4 ) = R2 (W4 ) = f (2) + 1. We may assume that k ≥ 3. Let G := Kf (k)+1 and let τ : E(G) − → [k] be any Gallai k-coloring of G. Suppose that (G, τ ) does not contain any monochromatic copy of W4 . Then τ is bad. Among all complete graphs on f (k) + 1 vertices with a bad Gallai k-coloring, we choose (G, τ ) with k minimum. By the minimality of k, we have GRl (W4 ) ≤ f (l) + 1 for all l ∈ {2, . . . , k − 1}. Throughout the proof, for any Gallai-colored complete graph (H , σ ) and any subset A of V (H), we use (H [A], σ ) to denote (H [A], σA ), where σA is the restriction of σ on the edges of H [A]. We next prove several claims. Claim 1. Let A ⊆ V (G) and let τl be a Gallai l-coloring of G[A] with no monochromatic copy of W4 , where 3 ≤ l ≤ k. Let i be a color used by τl . If (G[A], τl ) has no monochromatic copy of P3 in color i, then |A| ≤ f (l − 1). Proof. Suppose |A| ≥ f (l − 1) + 1. By the minimality of k, GRl−1 (W4 ) ≤ f (l − 1) + 1. Then |A| ≥ GRl−1 (W4 ). We may assume that the color i is blue. Since l ≥ 3, let red be another color in [l]\{i}, and let τ ∗ be obtained from τl by replacing colors red and blue with a new color, say β ∈ / [l]. Then τ ∗ is a Gallai (l − 1)-coloring of G and so (G[A], τ ∗ ) contains a monochromatic copy of W := W4 because |A| ≥ GRl−1 (W4 ). Since (G[A], τl ) contains no monochromatic W4 , it follows that W must be a monochromatic W4 in color β . For the remainder of the proof of this claim, we use W4 = (u1 , u2 , u3 , u4 ; u0 ) to denote a wheel on 5 vertices, where W4 \ u0 is a cycle with vertices u1 , u2 , u3 , u4 in order, and u0 is complete to {u1 , u2 , u3 , u4 }. Let W = (u1 , u2 , u3 , u4 ; u0 ). Then W has at most two independent blue edges under τl , because (G[A], τl ) has no blue P3 . Next, we show that W [{u1 , u2 , u3 , u4 }] is a red C4 under τl . Without loss of generality, suppose that u1 u2 is blue. Since (G[A], τl ) has no blue P3 , we see that u1 u4 , u1 u0 , u2 u3 , u2 u0 must be red. We further observe that u1 u3 and u2 u4 must be red because (G[A], τl ) has no rainbow triangle. It follows that {u1 , u2 } is red-complete to {u3 , u4 , u0 } in G under τl . Note that one of u0 u3 and u0 u4 is red since (G[A], τl ) has no blue P3 . We may further assume that u0 u3 is red under τl . But then we obtain a red W4 = (u1 , u4 , u2 , u3 ; u0 ) under τl when u0 u4 is red, and a red W4 = (u0 , u2 , u4 , u1 ; u3 ) under τl when u0 u4 is blue, contrary to the fact that (G, τl ) has no monochromatic copy of W4 . This proves that W [{u1 , u2 , u3 , u4 }] is a red C4 under τl . Since (G[A], τl ) has no blue P3 , we see that exactly one of the edges u0 u1 , u0 u2 , u0 u3 , u0 u4 , say u0 u1 is blue under τl . But then we obtain a red W4 = (u1 , u2 , u0 , u4 ; u3 ) under τl , contrary to the fact that (G, τl ) has no monochromatic copy of W4 . ■ Claim 2. Let A ⊆ V (G) and let i, j ∈ [k] be two distinct colors. If (G[A], τ ) has no monochromatic copy of P3 in color i or in color j, then |A| ≤ f (k − 2). Proof. We may assume that color i is red and color j is blue. Then (G[A], τ ) has neither red nor blue copy of P3 . Let τ ∗ be obtained from τ by replacing colors red and blue with a new color, say β ∈ / [k]. Then τ ∗ is a Gallai (k − 1)-coloring ∗ of G. We claim that (G[A], τ ) has no monochromatic copy of P3 in color β . Suppose not. Let x1 , x2 , x3 ∈ A be such that x1 x2 , x2 x3 are colored by color β under τ ∗ . Since (G[A], τ ) has neither red nor blue P3 , we may assume that x1 x2 is colored blue and x2 x3 is colored red under τ . Then x1 x3 is colored neither red nor blue under τ . But then (G[{x1 , x2 , x3 }], τ ) is a rainbow triangle, a contradiction. Thus (G[A], τ ∗ ) has no monochromatic copy of P3 in color β , as claimed. Then (G[A], τ ∗ ) has no monochromatic copy of W4 . By Claim 1 applied to A, τ ∗ and color β , we have |A| ≤ f (k − 2). ■
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Claim 3. Let A ⊆ V (G) and let x, y ∈ V (G) \ A be distinct. If all the edges between {x, y} and A are colored the same color, say color i ∈ [k], under τ , then (G[A], τ ) contains no monochromatic copy of P3 in color i. Proof. We may assume that the color i is blue. Suppose (G[A], τ ) contains a blue copy of P3 with vertices, say v1 , v2 , v3 in order. Then we obtain a blue W4 = (v1 , x, v3 , y; v2 ), which is a contradiction. ■ Let x1 , x2 , . . . , xm ∈ V (G) be a maximum sequence of vertices chosen as follows: for each j ∈ [m], all edges between xj and V (G) \ {x1 , x2 , . . . , xj } are colored by the same color under τ . Let X := {x1 , x2 , . . . , xm }. Note that X is possibly empty. For each xj ∈ X , let τ (xj ) be the unique color on the edges between xj and V (G) \ {x1 , x2 , . . . , xj }. Claim 4. τ (xi ) ̸ = τ (xj ) for all i, j ∈ [m] with i ̸ = j. Proof. Suppose there exist i, j ∈ [m] with i ̸ = j such that τ (xi ) = τ (xj ). We may assume that xj is the first vertex in the sequence x1 , . . . , xm such that τ (xj ) = τ (xi ) for some i ∈ [m] with i < j. We may further assume that the color τ (xi ) is blue. By the pigeonhole principle, j ≤ k + 1. Let A := V (G) \ {x1 , x2 , . . . , xj }. Then all the edges between {xi , xj } and A are colored blue under τ . By Claim 3, (G[A], τ ) has no blue P3 . By Claim 1, |A| ≤ f (k − 1). Then |G| ≤ f (k − 1) + k + 1 < f (k) + 1, which is impossible. ■ By Claim 4, |X | ≤ k. Note that G \ X has no monochromatic copy of W4 under τ . Consider a Gallai-partition of G \ X , as given in Theorem 1.2, with parts V1 , V2 , . . . , Vp , where p ≥ 2. We may assume that |V1 | ≥ |V2 | ≥ · · · ≥ |Vp |. Let R be the reduced graph of G \ X with vertices a1 , . . . , ap . By Theorem 1.2, we may assume that every edge of R is colored red or blue. Note that any monochromatic copy of W4 in R would yield a monochromatic copy of W4 in G \ X . Thus R has neither red nor blue W4 . Since R2 (W4 ) = 15, we see that p ≤ 14. Then |V1 | ≥ 2 because |G \ X | ≥ 26 for all k ≥ 3. Let p
B := {v ∈ ∪i=2 Vi | v is blue-complete to V1 under τ } and p
R := {v ∈ ∪i=2 Vi | v is red-complete to V1 under τ }. Then |B| + |R| + |V1 | = |G| − |X |. We may assume that |B| ≤ |R|. Suppose |R| = 1. Then |B| ≤ 1. We may assume that the only edge between R and B is red if |B| = 1. Then all the edges between the vertex in R and V (G) \ (X ∪ R) are colored red under τ , contrary to the maximality of m when choosing x1 , . . . , xm . Thus |R| ≥ 2. By Claim 3, neither (G[V1 ], τ ) nor (G[R], τ ) contains a red copy of P3 . Note that (G[V1 ∪ R], τ ) has a red copy of C4 . Thus no vertex in X is red-complete to V (G) \ X under τ . Claim 5. (G[B], τ ) contains a red copy of P3 . Furthermore, |V1 ∪ X | ≤ f (k − 2). Proof. Suppose (G[B], τ ) has no red P3 . Since no vertex in X is red-complete to V (G) \ X under τ and (G, τ ) has no rainbow triangle, we see that (G[X ], τ ) has no red edge, and no edge between X and V (G) \ X is colored red under τ . Then (G[B ∪ V1 ∪ X ], τ ) has no red P3 , because (G[V1 ], τ ) has no red P3 . Note that (G[R], τ ) has no red P3 . By Claim 1, |R| ≤ f (k − 1) and |B ∪ V1 ∪ X | ≤ f (k − 1). By (∗), 2f (k − 1) ≤ f (k) for all k ≥ 3. But then |G| = |R| + |B ∪ V1 ∪ X | ≤ 2f (k − 1) ≤ f (k), which contradicts the fact that |G| = f (k) + 1. Thus (G[B], τ ) contains a red copy of P3 and so |B| ≥ 3. If some vertex x ∈ X is blue-complete to V (G) \ X under τ , then (G, τ ) contains a blue W4 = (v1 , b1 , v2 , b2 ; x), where v1 , v2 ∈ V1 , and b1 , b2 ∈ B. Thus no vertex in X is blue-complete to V (G) \ X under τ . It follows that (G[X ], τ ) has no blue edge, and no edge between X and V (G) \ X is colored blue under τ . By Claim 3, (G[V1 ], τ ) has no blue P3 . Thus (G[V1 ∪ X ], τ ) contains neither red nor blue P3 . By Claim 2, |V1 ∪ X | ≤ f (k − 2), as desired. ■ Claim 6. |V3 | ≤ 1. Proof. Suppose |V1 | ≥ |V2 | ≥ |V3 | ≥ 2. Recall that a1 , a2 , a3 are the corresponding vertices of V1 , V2 , V3 in the reduced graph R. If R[{a1 , a2 , a3 }] is a monochromatic triangle, say red, then (G, τ ) contains a red W4 = (v1 , v3 , v2 , v4 ; v5 ), where v1 , v2 ∈ V1 , v3 , v4 ∈ V2 and v5 ∈ V3 . Thus R[{a1 , a2 , a3 }] is not a monochromatic triangle. Let A1 , A2 , A3 be a permutation of V1 , V2 , V3 such that A2 is, say, blue-complete, to A1 ∪ A3 in G. Then A1 must be red-complete to A3 . Let A := V (G) \ (A1 ∪ A2 ∪ A3 ∪ X ). Note that there is a red C4 using edges between A1 and A3 , and there is a blue C4 using edges between A1 and A2 . Since (G, τ ) has neither red nor blue W4 , we see that no vertex in A is red-complete to A1 ∪ A3 , and no vertex in A is blue-complete to A1 ∪ A2 or A2 ∪ A3 . This implies that A must be red-complete to A2 . Note that every vertex in A is blue-complete to either A1 or A3 . Let A∗ := {v ∈ A | v is blue-complete to A1 under τ }. Then A \ A∗ is blue-complete to A3 under τ . Note that A1 is blue-complete to A∗ and A2 is red-complete to A∗ . By Claim 3, (G[A∗ ], τ ) has neither red nor blue P3 . By Claim 2, |A∗ | ≤ f (k − 2). Similarly, |A \ A∗ | ≤ f (k − 2). By Claim 5, |Ai | ≤ |V1 | ≤ f (k − 2) − |X | for all i ∈ {1, 2, 3}. By (∗), 5f (k − 2) ≤ f (k) for all k ≥ 3. But then
|G| = |A∗ | + |A \ A∗ | + |A1 | + |A2 | + |A3 | + |X | ≤ 5f (k − 2) − 2|X | ≤ f (k), contrary to the fact that |G| = f (k) + 1.
■
Z.-X. Song, B. Wei, F. Zhang et al. / Discrete Mathematics 343 (2020) 111725
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By Claim 6, |V3 | ≤ 1. By Claim 5, |V1 ∪ X | ≤ f (k − 2) and so |V2 | ≤ f (k − 2). By (∗) again, 5f (k − 2) ≤ f (k) for all k ≥ 3. But then
|G| = |V1 ∪ X | + |V2 | + |V3 | + · · · + |Vp | ≤ 2f (k − 2) + (p − 2) ≤ 2f (k − 2) + 12 ≤ 5f (k − 2) ≤ f (k), because p ≤ 14 and f (k − 2) ≥ 4, contrary to the fact that |G| = f (k) + 1. This completes the proof of Theorem 1.6. ■ Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors thank both referees for their careful reading and helpful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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Note
A note on Gallai–Ramsey number of even wheels ∗
Zi-Xia Song a ,1 , Bing Wei b , Fangfang Zhang c,a , ,2 , Qinghong Zhao b a
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Department of Mathematics, University of Mississippi, University, MS 38677, USA c Department of Mathematics, Nanjing University, Nanjing 210093, PR China b
article
info
Article history: Received 28 January 2019 Received in revised form 30 October 2019 Accepted 31 October 2019 Available online xxxx Keywords: Gallai coloring Gallai-Ramsey number Rainbow triangle
a b s t r a c t A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses k colors. Given a graph H and an integer k ≥ 1, the Gallai–Ramsey number GRk (H) of H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. Let Wn denote a wheel on n + 1 vertices. In this note, we study Gallai–Ramsey number of W2n and completely determine the exact value of GRk (W4 ) for all k ≥ 2. © 2019 Elsevier B.V. All rights reserved.
1. Introduction All graphs in this paper are finite, simple and undirected. Given a graph G and a set S ⊆ V (G), we use |G| to denote the number of vertices of G, and G[S ] to denote the subgraph of G obtained from G by deleting all vertices in V (G) \ S. For two disjoint sets A, B ⊆ V (G), A is complete to B in G if every vertex in A is adjacent to all vertices in B. We use Kn , Cn , Pn to denote the complete graph, cycle, and path on n vertices, respectively; and Wn to denote a wheel on n + 1 vertices. For any positive integer k, we write [k] for the set {1, . . . , k}. We use the convention ‘‘S :=’’ to mean that S is defined to be the right-hand side of the relation. Given an integer k ≥ 1 and graphs H1 , . . . , Hk , the classical Ramsey number R(H1 , . . . , Hk ) is the least integer N such that every k-coloring of the edges of KN contains a monochromatic copy of Hi in color i for some i ∈ [k]. When H = H1 = · · · = Hk , we simply write Rk (H). Ramsey numbers are notoriously difficult to compute in general. Very little is known for Ramsey numbers of wheels and the exact value of R2 (Wn ) is only known for n ∈ {3, 4, 5} (see [7,11,14]). In this paper, we study Ramsey number of wheels in Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles (that is, a triangle with all its edges colored differently). Gallai colorings naturally arise in several areas including: information theory [15]; the study of partially ordered sets, as in Gallai’s original paper [10] (his result was restated in [13] in the terminology of graphs); and the study of perfect graphs [4]. More information on this topic can be found in [8,9]. A Gallai k-coloring is a Gallai coloring that uses at most k colors. We use (G, τ ) to denote a Gallai k-colored complete graph if G is a complete graph and τ : E(G) → [k] is a Gallai k-coloring. Given an integer k ≥ 1 and graphs H1 , . . . , Hk , the Gallai-Ramsey number GR(H1 , . . . , Hk ) is the least integer N such that every (KN , τ ) contains a monochromatic copy of Hi in color i for some i ∈ [k]. When H = H1 = · · · = Hk , we simply write GRk (H). Clearly, GRk (H) ≤ Rk (H) for all k ≥ 1 and GR(H1 , H2 ) = R(H1 , H2 ). In 2010, Gyárfás, Sárközy, Sebő and Selkow [12] proved the general behavior of GRk (H). ∗ Corresponding author at: Department of Mathematics, Nanjing University, Nanjing 210093, PR China. E-mail addresses: [email protected] (Z.-X. Song), [email protected] (F. Zhang). 1 Supported by the National Science Foundation, USA under Grant No. DMS-1854903. 2 The work is done while the third author was studying at the University of Central Florida as a visiting student, supported by the Chinese Scholarship Council, PR China. https://doi.org/10.1016/j.disc.2019.111725 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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Theorem 1.1 ([12]). Let H be a fixed graph with no isolated vertices and let k ≥ 1 be an integer. Then GRk (H) is exponential in k if H is not bipartite, linear in k if H is bipartite but not a star, and constant (does not depend on k) when H is a star. It turns out that for some graphs H (e.g., when H = C3 ), GRk (H) behaves nicely, while the order of magnitude of Rk (H) seems hopelessly difficult to determine. It is worth noting that finding exact values of GRk (H) is far from trivial, even when |H | is small. The following structural result of Gallai [10] is crucial in determining the exact value of Gallai-Ramsey number of a graph H. Theorem 1.2 ([10]). For any Gallai coloring τ of a complete graph G with |G| ≥ 2, V (G) can be partitioned into nonempty sets V1 , V2 , . . . , Vp with p > 1 so that at most two colors are used on the edges in E(G) \ (E(V1 ) ∪ · · · ∪ E(Vp )) and only one color is used on the edges between any fixed pair (Vi , Vj ) under τ , where E(Vi ) denotes the set of edges in G[Vi ] for all i ∈ [p]. The partition given in Theorem 1.2 is a Gallai partition of the complete graph G under τ . Given a Gallai partition V1 , . . . , Vp of the complete graph G under τ , let vi ∈ Vi for all i ∈ [p] and let R := G[{v1 , . . . , vp }]. Then R is the reduced graph of G corresponding to the given Gallai partition under τ . By Theorem 1.2, all edges in R are colored by at most two colors under τ . One can see that any monochromatic copy of H in R will result in a monochromatic copy of H in G under τ . It is not surprising that Gallai-Ramsey number GRk (H) is closely related to the classical Ramsey number R2 (H). The following is a conjecture of Fox, Grinshpun and Pach [8] on GRk (H) when H is a complete graph. Conjecture 1.3 ([8]). For all k ≥ 1 and t ≥ 3, (R2 (Kt ) − 1)k/2 + 1
{ GRk (Kt ) =
(t − 1)(R2 (Kt ) − 1)
if k is even
(k−1)/2
+ 1 if k is odd.
The first case t = 3 of Conjecture 1.3 follows directly from a result of Chung and Graham [6] from 1983, and a simpler proof can be found in [12]. Recently, the case t = 4 of Conjecture 1.3 was proved in [18]. In this note, we study Gallai–Ramsey number of even wheels. Recall that (G, τ ) denotes a Gallai k-colored complete graph if G is a complete graph and τ : E(G) → [k] is a Gallai k-coloring. For any A, B ⊆ V (G), if all the edges between A and B are colored the same color, say blue, under τ , we say that A is blue-complete to B in (G, τ ). We simply say a is blue-complete to B in (G, τ ) when A = {a}. We first prove the following general lower bound for GRk (W2n ) for all k ≥ 2 and n ≥ 2. Proposition 1.4.
For all n ≥ 2 and k ≥ 2, (R2 (W2n ) − 1) · 5(k−2)/2 + 1
{ GRk (W2n ) ≥
if k is even
2(R2 (W2n ) − 1) · 5(k−3)/2 + 1 if k is odd.
Proof. Let n ≥ 2 be given. For all k ≥ 2, let
{ f (k) :=
(R2 (W2n ) − 1) · 5(k−2)/2
if k is even
2(R2 (W2n ) − 1) · 5(k−3)/2
if k is odd.
For all k ≥ 2, we next construct (Gk , τk ) recursively as follows, where Gk is a complete graph on f (k) vertices and τk : E(Gk ) → [k] is a Gallai k-coloring such that (Gk , τk ) has no monochromatic copy of W2n . For k = 2, let m := f (2) = R2 (W2n ) − 1 and G2 := Km . By the definition of R2 (W2n ), there exists τ2 : E(G2 ) → {1, 2} such that G2 has no monochromatic copy of W2n under τ2 . Then (G2 , τ2 ) is a Gallai 2-colored complete graph on f (2) vertices with no monochromatic copy of W2n , as desired. When k ≥ 3 is even, let (Gk−2 , τk−2 ) be the construction on f (k − 2) vertices with colors in [k − 2] and let (H , c) := (K5 , c) be such that edges of H are colored by colors k and k − 1 with no monochromatic copy of K3 . Let (Gk , τk ) be obtained from (H , c) by replacing each vertex of H with a copy of (Gk−2 , τk−2 ) such that for every uv ∈ E(H), all the edges between the corresponding copies of (Gk−2 , τk−2 ) for u and v are colored by the color c(uv ). Since (Gk−2 , τk−2 ) is a Gallai (k − 2)-colored complete graph with no monochromatic copy of W2n , we see that τk is indeed a Gallai k-coloring and (Gk , τk ) has no monochromatic copy of W2n . When k ≥ 3 is odd, let (Gk−1 , τk−1 ) be the construction on f (k − 1) vertices with colors in [k − 1] and let (Gk , τk ) be the join of two disjoint copies of (Gk−1 , τk−1 ), with all the new edges colored by the color k. In both cases, (Gk , τk ) has no rainbow triangle and no monochromatic copy of W2n . Hence, GRk (W2n ) ≥ f (k) + 1, as desired. ■ We will make use of the following result of Hendry [14] on the exact value of R2 (W4 ). Theorem 1.5 ([14]). R2 (W4 ) = 15. The main result in this note is Theorem 1.6, which establishes the exact value of GRk (W4 ) for all k ≥ 2. Theorem 1.6. For all k ≥ 2,
{ GRk (W4 ) =
14 · 5
k−2 2
28 · 5
k−3 2
+ 1, + 1,
if k is even if k is odd.
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With the support of Theorem 1.6, it seems that the lower bound given in Proposition 1.4 is also the desired upper bound for GRk (W2n ). We propose the following conjecture. Conjecture 1.7. For all k ≥ 2 and n ≥ 2,
{ GRk (W2n ) =
(R2 (W2n ) − 1) · 5
k−2 2
2(R2 (W2n ) − 1) · 5
k−3 2
+ 1, + 1,
if k is even if k is odd.
It is worth noting that very recently, Gallai-Ramsey number of some other graphs with at most five vertices were studied by Li and Wang [17]. More recent work on Gallai-Ramsey numbers of cycles can be found in [1–3,5,16,19,20]. 2. Proof of Theorem 1.6 Let f (1) := 4 and for each integer s ≥ 2, let
{ f (s) :=
14 · 5
s−2 2
28 · 5
s−3 2
, ,
if s is even if s is odd.
Then for all s ≥ 3,
{ f (s) ≥
2f (s − 1) 5f (s − 2).
(⋆)
By Proposition 1.4, it suffices to show that for all k ≥ 2, GRk (W4 ) ≤ f (k) + 1. By Theorem 1.5, R2 (W4 ) = 15 and so GR2 (W4 ) = R2 (W4 ) = f (2) + 1. We may assume that k ≥ 3. Let G := Kf (k)+1 and let τ : E(G) − → [k] be any Gallai k-coloring of G. Suppose that (G, τ ) does not contain any monochromatic copy of W4 . Then τ is bad. Among all complete graphs on f (k) + 1 vertices with a bad Gallai k-coloring, we choose (G, τ ) with k minimum. By the minimality of k, we have GRl (W4 ) ≤ f (l) + 1 for all l ∈ {2, . . . , k − 1}. Throughout the proof, for any Gallai-colored complete graph (H , σ ) and any subset A of V (H), we use (H [A], σ ) to denote (H [A], σA ), where σA is the restriction of σ on the edges of H [A]. We next prove several claims. Claim 1. Let A ⊆ V (G) and let τl be a Gallai l-coloring of G[A] with no monochromatic copy of W4 , where 3 ≤ l ≤ k. Let i be a color used by τl . If (G[A], τl ) has no monochromatic copy of P3 in color i, then |A| ≤ f (l − 1). Proof. Suppose |A| ≥ f (l − 1) + 1. By the minimality of k, GRl−1 (W4 ) ≤ f (l − 1) + 1. Then |A| ≥ GRl−1 (W4 ). We may assume that the color i is blue. Since l ≥ 3, let red be another color in [l]\{i}, and let τ ∗ be obtained from τl by replacing colors red and blue with a new color, say β ∈ / [l]. Then τ ∗ is a Gallai (l − 1)-coloring of G and so (G[A], τ ∗ ) contains a monochromatic copy of W := W4 because |A| ≥ GRl−1 (W4 ). Since (G[A], τl ) contains no monochromatic W4 , it follows that W must be a monochromatic W4 in color β . For the remainder of the proof of this claim, we use W4 = (u1 , u2 , u3 , u4 ; u0 ) to denote a wheel on 5 vertices, where W4 \ u0 is a cycle with vertices u1 , u2 , u3 , u4 in order, and u0 is complete to {u1 , u2 , u3 , u4 }. Let W = (u1 , u2 , u3 , u4 ; u0 ). Then W has at most two independent blue edges under τl , because (G[A], τl ) has no blue P3 . Next, we show that W [{u1 , u2 , u3 , u4 }] is a red C4 under τl . Without loss of generality, suppose that u1 u2 is blue. Since (G[A], τl ) has no blue P3 , we see that u1 u4 , u1 u0 , u2 u3 , u2 u0 must be red. We further observe that u1 u3 and u2 u4 must be red because (G[A], τl ) has no rainbow triangle. It follows that {u1 , u2 } is red-complete to {u3 , u4 , u0 } in G under τl . Note that one of u0 u3 and u0 u4 is red since (G[A], τl ) has no blue P3 . We may further assume that u0 u3 is red under τl . But then we obtain a red W4 = (u1 , u4 , u2 , u3 ; u0 ) under τl when u0 u4 is red, and a red W4 = (u0 , u2 , u4 , u1 ; u3 ) under τl when u0 u4 is blue, contrary to the fact that (G, τl ) has no monochromatic copy of W4 . This proves that W [{u1 , u2 , u3 , u4 }] is a red C4 under τl . Since (G[A], τl ) has no blue P3 , we see that exactly one of the edges u0 u1 , u0 u2 , u0 u3 , u0 u4 , say u0 u1 is blue under τl . But then we obtain a red W4 = (u1 , u2 , u0 , u4 ; u3 ) under τl , contrary to the fact that (G, τl ) has no monochromatic copy of W4 . ■ Claim 2. Let A ⊆ V (G) and let i, j ∈ [k] be two distinct colors. If (G[A], τ ) has no monochromatic copy of P3 in color i or in color j, then |A| ≤ f (k − 2). Proof. We may assume that color i is red and color j is blue. Then (G[A], τ ) has neither red nor blue copy of P3 . Let τ ∗ be obtained from τ by replacing colors red and blue with a new color, say β ∈ / [k]. Then τ ∗ is a Gallai (k − 1)-coloring ∗ of G. We claim that (G[A], τ ) has no monochromatic copy of P3 in color β . Suppose not. Let x1 , x2 , x3 ∈ A be such that x1 x2 , x2 x3 are colored by color β under τ ∗ . Since (G[A], τ ) has neither red nor blue P3 , we may assume that x1 x2 is colored blue and x2 x3 is colored red under τ . Then x1 x3 is colored neither red nor blue under τ . But then (G[{x1 , x2 , x3 }], τ ) is a rainbow triangle, a contradiction. Thus (G[A], τ ∗ ) has no monochromatic copy of P3 in color β , as claimed. Then (G[A], τ ∗ ) has no monochromatic copy of W4 . By Claim 1 applied to A, τ ∗ and color β , we have |A| ≤ f (k − 2). ■
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Claim 3. Let A ⊆ V (G) and let x, y ∈ V (G) \ A be distinct. If all the edges between {x, y} and A are colored the same color, say color i ∈ [k], under τ , then (G[A], τ ) contains no monochromatic copy of P3 in color i. Proof. We may assume that the color i is blue. Suppose (G[A], τ ) contains a blue copy of P3 with vertices, say v1 , v2 , v3 in order. Then we obtain a blue W4 = (v1 , x, v3 , y; v2 ), which is a contradiction. ■ Let x1 , x2 , . . . , xm ∈ V (G) be a maximum sequence of vertices chosen as follows: for each j ∈ [m], all edges between xj and V (G) \ {x1 , x2 , . . . , xj } are colored by the same color under τ . Let X := {x1 , x2 , . . . , xm }. Note that X is possibly empty. For each xj ∈ X , let τ (xj ) be the unique color on the edges between xj and V (G) \ {x1 , x2 , . . . , xj }. Claim 4. τ (xi ) ̸ = τ (xj ) for all i, j ∈ [m] with i ̸ = j. Proof. Suppose there exist i, j ∈ [m] with i ̸ = j such that τ (xi ) = τ (xj ). We may assume that xj is the first vertex in the sequence x1 , . . . , xm such that τ (xj ) = τ (xi ) for some i ∈ [m] with i < j. We may further assume that the color τ (xi ) is blue. By the pigeonhole principle, j ≤ k + 1. Let A := V (G) \ {x1 , x2 , . . . , xj }. Then all the edges between {xi , xj } and A are colored blue under τ . By Claim 3, (G[A], τ ) has no blue P3 . By Claim 1, |A| ≤ f (k − 1). Then |G| ≤ f (k − 1) + k + 1 < f (k) + 1, which is impossible. ■ By Claim 4, |X | ≤ k. Note that G \ X has no monochromatic copy of W4 under τ . Consider a Gallai-partition of G \ X , as given in Theorem 1.2, with parts V1 , V2 , . . . , Vp , where p ≥ 2. We may assume that |V1 | ≥ |V2 | ≥ · · · ≥ |Vp |. Let R be the reduced graph of G \ X with vertices a1 , . . . , ap . By Theorem 1.2, we may assume that every edge of R is colored red or blue. Note that any monochromatic copy of W4 in R would yield a monochromatic copy of W4 in G \ X . Thus R has neither red nor blue W4 . Since R2 (W4 ) = 15, we see that p ≤ 14. Then |V1 | ≥ 2 because |G \ X | ≥ 26 for all k ≥ 3. Let p
B := {v ∈ ∪i=2 Vi | v is blue-complete to V1 under τ } and p
R := {v ∈ ∪i=2 Vi | v is red-complete to V1 under τ }. Then |B| + |R| + |V1 | = |G| − |X |. We may assume that |B| ≤ |R|. Suppose |R| = 1. Then |B| ≤ 1. We may assume that the only edge between R and B is red if |B| = 1. Then all the edges between the vertex in R and V (G) \ (X ∪ R) are colored red under τ , contrary to the maximality of m when choosing x1 , . . . , xm . Thus |R| ≥ 2. By Claim 3, neither (G[V1 ], τ ) nor (G[R], τ ) contains a red copy of P3 . Note that (G[V1 ∪ R], τ ) has a red copy of C4 . Thus no vertex in X is red-complete to V (G) \ X under τ . Claim 5. (G[B], τ ) contains a red copy of P3 . Furthermore, |V1 ∪ X | ≤ f (k − 2). Proof. Suppose (G[B], τ ) has no red P3 . Since no vertex in X is red-complete to V (G) \ X under τ and (G, τ ) has no rainbow triangle, we see that (G[X ], τ ) has no red edge, and no edge between X and V (G) \ X is colored red under τ . Then (G[B ∪ V1 ∪ X ], τ ) has no red P3 , because (G[V1 ], τ ) has no red P3 . Note that (G[R], τ ) has no red P3 . By Claim 1, |R| ≤ f (k − 1) and |B ∪ V1 ∪ X | ≤ f (k − 1). By (∗), 2f (k − 1) ≤ f (k) for all k ≥ 3. But then |G| = |R| + |B ∪ V1 ∪ X | ≤ 2f (k − 1) ≤ f (k), which contradicts the fact that |G| = f (k) + 1. Thus (G[B], τ ) contains a red copy of P3 and so |B| ≥ 3. If some vertex x ∈ X is blue-complete to V (G) \ X under τ , then (G, τ ) contains a blue W4 = (v1 , b1 , v2 , b2 ; x), where v1 , v2 ∈ V1 , and b1 , b2 ∈ B. Thus no vertex in X is blue-complete to V (G) \ X under τ . It follows that (G[X ], τ ) has no blue edge, and no edge between X and V (G) \ X is colored blue under τ . By Claim 3, (G[V1 ], τ ) has no blue P3 . Thus (G[V1 ∪ X ], τ ) contains neither red nor blue P3 . By Claim 2, |V1 ∪ X | ≤ f (k − 2), as desired. ■ Claim 6. |V3 | ≤ 1. Proof. Suppose |V1 | ≥ |V2 | ≥ |V3 | ≥ 2. Recall that a1 , a2 , a3 are the corresponding vertices of V1 , V2 , V3 in the reduced graph R. If R[{a1 , a2 , a3 }] is a monochromatic triangle, say red, then (G, τ ) contains a red W4 = (v1 , v3 , v2 , v4 ; v5 ), where v1 , v2 ∈ V1 , v3 , v4 ∈ V2 and v5 ∈ V3 . Thus R[{a1 , a2 , a3 }] is not a monochromatic triangle. Let A1 , A2 , A3 be a permutation of V1 , V2 , V3 such that A2 is, say, blue-complete, to A1 ∪ A3 in G. Then A1 must be red-complete to A3 . Let A := V (G) \ (A1 ∪ A2 ∪ A3 ∪ X ). Note that there is a red C4 using edges between A1 and A3 , and there is a blue C4 using edges between A1 and A2 . Since (G, τ ) has neither red nor blue W4 , we see that no vertex in A is red-complete to A1 ∪ A3 , and no vertex in A is blue-complete to A1 ∪ A2 or A2 ∪ A3 . This implies that A must be red-complete to A2 . Note that every vertex in A is blue-complete to either A1 or A3 . Let A∗ := {v ∈ A | v is blue-complete to A1 under τ }. Then A \ A∗ is blue-complete to A3 under τ . Note that A1 is blue-complete to A∗ and A2 is red-complete to A∗ . By Claim 3, (G[A∗ ], τ ) has neither red nor blue P3 . By Claim 2, |A∗ | ≤ f (k − 2). Similarly, |A \ A∗ | ≤ f (k − 2). By Claim 5, |Ai | ≤ |V1 | ≤ f (k − 2) − |X | for all i ∈ {1, 2, 3}. By (∗), 5f (k − 2) ≤ f (k) for all k ≥ 3. But then
|G| = |A∗ | + |A \ A∗ | + |A1 | + |A2 | + |A3 | + |X | ≤ 5f (k − 2) − 2|X | ≤ f (k), contrary to the fact that |G| = f (k) + 1.
■
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By Claim 6, |V3 | ≤ 1. By Claim 5, |V1 ∪ X | ≤ f (k − 2) and so |V2 | ≤ f (k − 2). By (∗) again, 5f (k − 2) ≤ f (k) for all k ≥ 3. But then
|G| = |V1 ∪ X | + |V2 | + |V3 | + · · · + |Vp | ≤ 2f (k − 2) + (p − 2) ≤ 2f (k − 2) + 12 ≤ 5f (k − 2) ≤ f (k), because p ≤ 14 and f (k − 2) ≥ 4, contrary to the fact that |G| = f (k) + 1. This completes the proof of Theorem 1.6. ■ Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors thank both referees for their careful reading and helpful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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