On repetition thresholds of caterpillars and trees of bounded degree

The repetition threshold is the smallest real number α such that there exists an inﬁnite word over a k -letter alphabet that avoids repetition of exponent strictly greater than α . This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree 3. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.


Introduction
A word w of length |w|=r over an alphabet A is a sequence w 1 . . . w r of r letters, i.e. r elements of A. A prefix of a word w = w 1 . . . w r is a word p = w 1 . . . w s , for some s r.
A repetition in a word w is a pair of words p (called the period ) and e (called the excess) such that pe is a factor of w, p is non-empty, and e is a prefix of pe. The exponent of a repetition pe is exp(pe) = |pe| |p| . A β-repetition is a repetition of exponent β. A word is α + -free (resp. α-free) if it contains no β-repetition such that β > α (resp. β α).
Given k 2, Dejean [7] defined the repetition threshold RT(k) for k letters as the smallest α such that there exists an infinite α + -free word over a k-letter alphabet.
Dejean initiated the study of RT(k) in 1972 for k = 2 and k = 3. Her work was followed by a series of papers which determine the exact value of RT(k) for any k 2.
The notions of α-free word and α + -free word have been generalized to graphs. A graph G is determined by a set of vertices V (G) and a set of edges E(G). A mapping c : V (G) → {1, . . . , k} is a k-coloring of G. A sequence of colors on a non-intersecting path in a k-colored graph G is called a factor. A coloring is said to be α + -free (resp. α-free) if every factor is α + -free (resp. α-free).
The notion of repetition threshold can be generalized to graphs as follows. Given a graph G and k colors, When considering the repetition threshold over a whole class of graphs G, it is defined as In the remainder of this paper, P, C, S, T , T k , CP, and CP k respectively denote the classes of paths, cycles, subdivisions 1 , trees, trees of maximum degree k, caterpillars and caterpillars of maximum degree k.
Since α + -free words are closed under reversal, the repetition thresholds for paths are clearly defined as RT(k, P) = RT(k), and thus Theorem 1 completely determines RT(k, P).
In 2004, Aberkane and Currie [1] initiated the study of the repetition threshold of cycles for 2 letters. Another result of Currie [3] on ternary circular square-free word allows to determine the repetition threshold of cycles for 3 letters. In 2012, Gorbunova [8] determined the repetition threshold of cycles for k 6 letters.  (iii) RT(k, C) = 1 + 1 k /2 , for k 6 [8].
Gorbunova [8] also conjectured that RT(4, C) = 3 2 and RT(4, C) = 4 3 . For the classes of graph subdivisions and trees, the bounds are completely determined [11].   (iii) RT(k, T ) = 3 2 , for k 4. In this paper, we continue the study of repetition thresholds of trees under additional assumptions. In particular, we completely determine the repetition thresholds for caterpillars of maximum degree 3 (Theorems 5 to 7 and 11) and for caterpillars of unbounded maximum degree (Theorems 5 and 6) for every alphabet of size k 2. We determine the repetition thresholds for trees of maximum degree 3 for every alphabet of size k ∈ {4, 5} (Theorem 12). We finally give a lower and an upper bound on the repetition threshold for trees of maximum degree 3 for every alphabet of size k 6 (Theorem 13). We summarize the results in Table 1 (shaded cells correspond to our results).

Caterpillars
A caterpillar is a tree such that the graph induced by the vertices of degree at least 2 is a path, which is called backbone.
Proof. First, we show that the repetition threshold is at least 3. Note that it suffices to prove it for the class of caterpillars with maximum degree 3. Suppose, to the contrary, that RT(2, CP 3 ) < 3. Then, the factor xxx is forbidden for any x ∈ A. Therefore, in any 3-free 2-coloring, every vertex colored with x has at most one neighbor colored with x. It follows that four consecutive backbone vertices of degree 3 cannot be colored xyxy for any x, y ∈ A, since the 3-repetition yxyxyx appears. The factor xyx is also forbidden. Indeed, xyx must extend to xxyxx on the backbone since xyxy is forbidden. Then, xxyxx must extend to yxxyxxy on the backbone since xxx is forbidden. Finally, yxxyxxy must extend to the 3-repetition xyxxyxxyx in the caterpillar. Thus, the binary word on the backbone must avoid xxx and xyx. So, this word must be (0011) ω which is not 3-free, a contradiction. Hence, RT(2, CP 3 ) 3. Now, consider a 2-coloring of an arbitrary caterpillar such that the backbone induces a 2 + -free word (which exists by Theorem 1(i)) and every pendent vertex gets the z x y x z y x z z y x Figure 1: After a factor xyx, the remaining colors are forced.
color distinct from the color of its neighbor. Clearly, this 2-coloring is 3 + -free, and so RT(2, CP) 3.
Proof. We start by proving RT(3, CP 3 ) 2. So, suppose, for a contradiction, that there is a 2-free 3-coloring for any caterpillar with maximum degree 3. In every 2-free 3-coloring, the factor xyx appears on the backbone, since otherwise the word on the backbone would be (012) ω which is not 2-free. Then, we have no choice to extend the factor xyx to the right (see Figure 1). This induces a 2-repetition yxzyxz. Now, we show that RT(3, CP) 2 by constructing a 2 + -free 3-coloring of an arbitrary caterpillar. Take a 2 + -free 2-coloring of the backbone (which exists by Theorem 1), and color the pendent vertices with the third color.  2 -free coloring, the vertices u 1 , u 2 , u 3 , v 2 must get distinct colors: say c(u 1 ) = x, c(u 2 ) = y, c(u 3 ) = z, c(v 2 ) = t. Either u 0 or u 4 must be colored with color t; w.l.o.g. assume c(u 4 ) = t. Then, either u 5 or v 4 must be colored by y, and we obtain the 5 3 -repetition tyzty.
(a) There exist 2η vertices at distance at most η from each other.
x 1 x 2 x 3 x k−2 Proof. Let η = k 2 . Suppose, to the contrary, that there exists a (1 + 1 η )-free k-coloring c for any caterpillar with maximum degree 3. Then, every two vertices at distance at most η must be colored differently. In caterpillars with maximum degree 3, we can have 2η vertices being pairwise at distance at most η (see Figure 2a). If k is odd, then 2η > k, and thus c is not (1 + 1 η )-free. If k is even, the vertices x i of Figure 2b necessarily get distinct colors, say x i gets color i. Then, we have c(y) ∈ {1, 3} and w.l.o.g. c(y) = 1. We also have 2 ∈ {c(z 1 ), c(z 2 )} and w.l.o.g. c(z 1 ) = 2. Then we obtain a 1 + 2 η+1 -repetition with excess c(y)c(z 1 ) = 12, a contradiction. Proof. We start from a right infinite 5 4 + -free word w = w 0 w 1 . . . on 5 letters. We associate to w its Pansiot code p such that p i = 0 if w i = w i+4 and p i = 1 otherwise, for every i 0 [12]. Let us construct a 4 3 + -free 5-coloring c of the infinite caterpillar such that every vertex on the backbone has exactly one pendant vertex . For every i 0, c[0 We check exhaustively that there exists no forbidden repetition of length at most 576 in the caterpillar. Now suppose for contradiction that there exists a repetition r of length n > 576 and exponent n d > 4 3 in the caterpillar. This implies that there exists a repetition of length n n − 2 and period of length d in the backbone. This repetition contains a repetition r consisting of full blocks of length 18 having length at least n − 2 × (18 − 1) n − 36 and period length d. Given n > 576 and n d > 4 3 , the repetition r has exponent at least n−36 d > 5 4 . The repetition r in the backbone implies a repetition of exponent greater than 5 4 in w, which is a contradiction.
Lemma 10. For every integer k 6, we have RT(k, CP 3 ) 1 + 1 k /2 . Proof. First notice that it suffices to construct colorings for odd k's, since 1 + 1 k /2 = 1 + 1 (k−1) /2 for k even. So, let k be odd and let η = k 2 . By Theorem 1(iv), we can color the vertices of the backbone by a (1 + 1 η ) + -free (η + 1)coloring. Then, it remains to color the pendant vertices: let us color them cyclically using the remaining k − (η + 1) = η − 2 unused colors (see Figure 3 for an example with k = 11). Clearly, the repetition which does contain a pendant vertex are (1 + 1 η ) + -free. Moreover, for a repetition containing a pendant vertex, the length of the excess is at most 1 and the period length is at least η. Thus, its exponent is at most η+1 η = 1 + 1 η . This shows that this k-coloring is (1 + 1 η ) + -free.
Lemmas 8 to 10 together imply the following theorem.
Theorem 11. For every integer k, with k 5, we have RT(k, Observe that for all k 4, we have RT(k, CP) = 3 2 . Indeed, caterpillars are trees and thus RT(k, CP) RT(k, T ) = 3 2 . On the other hand, we have RT(k, CP) RT(k, K 1,k ) = 3 2 (where K 1,k is the star of degree k).

Trees of maximum degree 3
The class of trees of maximum degree 3 is denoted by T 3 . Let T ∈ T 3 be the infinite embedded rooted tree whose vertices have degree 3, except the root which has degree 2.
Thus, every vertex of T has a left son and a right son. The level of a vertex of T is the distance to the root (the root has level 0). Since every tree of maximum degree 3 is a subgraph of T , we only consider T while proving that RT(k, T 3 ) α for some k and α. Let G ∈ T 3 be the graph depicted in Figure 4 and let v ∈ V (G) be the squared vertex. In every 5-coloring of G, at least two among the six vertices at distance 2 of v will get the same color. In every 3 2 -free 5-coloring, the distance between these two vertices is four. W.l.o.g., the two triangle vertices of Figure 4 are colored with the same color, say color 1. Then, we color the other vertices following the labels in alphabetical order (a vertex labelled x is called an x-vertex). The a-vertices have to get three distinct colors (and distinct from 1), say 2, 3, and 4. The b-vertices can only get colors 3 or 5 and they must have distinct colors in every 3 2 -free 5-coloring. This is the same for c-vertices. The d-vertex must then get color 5. Therefore the e-vertices can only get colors 2 or 4. The f -vertex must get color 4. Finally, the g-vertex cannot be colored without creating a forbidden factor. Thus RT(5, T 3 ) 3 2 , and that concludes the proof.

Conclusion
In this paper, we continued the study of repetition thresholds in colorings of various subclasses of trees. We completely determined the repetition thresholds for caterpillars and caterpillars of maximum degree 3, and presented some results for trees of maximum degree 3.
There are several open questions in the latter class for which it appears that more advanced methods of analysis should be developed. In particular, our bounds show that 3 RT (2, T 3 ) 7 2 and 2 RT (3, T 3 ) 3 , however, we have not been able to determine the exact bounds yet. Additionally, the repetition thresholds in trees of bounded degrees for alphabets of size at least 6 remain unknown.