α-Labelings and the Structure of Trees with Nonzero α-Deficit

We present theoretical and computational results onα-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of α-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero α-deficit, and prove an unexpected result on the α-deficit of trees with a vertex of large degree compared to the order of the tree.


Introduction
Let G = (V, E) be a graph such that n = |V |.A labeling of G is an injective function f from V to the set of integers {0, 1, . . ., |E|}.The induced label of each edge xy is |f (x) − f (y)|.If the resulting edge labels are distinct, then the labeling is said to be graceful.This notion was introduced by Rosa (1967) and many other graph labeling schemes have been proposed since (see Gallian (2009) for a regularly updated survey with more than one thousand references).
If T is a tree, then the function f is bijective and the set of possible vertex labels is {0, 1, . . ., n − 1}.A labeling of T is thus graceful if each integer from 1 to n−1 is assigned to an edge.The well known Graceful Tree Conjecture (i) states that every tree has a graceful labeling.Despite much effort, this conjecture is still open.However, some specific classes of trees are known to have a graceful labeling.Among them are: trees with at most 27 vertices (Aldred and McKay); trees with diameter at most 5 (Hrnčiar and Haviar (2001); Zhao (1989)); caterpillars (trees where the deletion of pendant vertices leaves a path, Rosa (1967)); symmetrical trees (rooted trees in which all the vertices at the same level have the same degree, Poljak and Sûra (1982); Bermond and Sotteau (1976)) and olive trees (rooted trees constructed from k branches such that the ith branch is a path of length i, Abhyankar and Bhat-Nayak (2000); Pastel and Raynaud (1978)).We refer to Alfalayleh et al. (2004) for a survey on known results towards proving the Graceful Tree Conjecture.
A labeling f is bipartite if there exists an integer k such that for each edge xy either f (x) ≤ k < f (y) or f (y) ≤ k < f (x).A graceful labeling that is bipartite is an α-labeling (Rosa and Širáň (1995)).Unlike graceful labelings, there are several examples of trees that do not have an α-labeling.The α-size α(T ) of a tree T is defined as the maximum number of distinct edge labels over all bipartite labelings of T .
Let α(n) be the smallest α-size among all trees with n vertices and α 3 (n) be the smallest α-size among all trees with maximum degree 3 and n vertices.Rosa and Širáň (1995) proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4. Bonnington and Širáň (1999) showed that α 3 (n) ≥ 5n/6 for all n ≥ 12.This last bound was improved by Brankovic et al. (2005b) to α 3 (n) ≥ 6n/7 − 1.In the case of trees with maximum degree 3 which have a perfect matching, Brankovic et al. (2005a) further improved the bound to α 3 (n) ≥ ((k − 1)n)/k − 1 where 2k is the lower bound on the number of vertices of a tree with maximum degree 3 having a perfect matching that does not admit an α-labeling.
In order to emphasize the fact that we are mainly interested in how close we can come to an α-labeling, instead of using the α-size of a tree T , we use the α-deficit α def defined as n − 1 − α(T ).
The parameter α def measures how far a tree is from having an α-labeling as it counts the minimum number of errors, that is, the minimum number of edge labels that are missing from the set of all possible labels.Trees with an α-labeling have deficit 0.
The main contributions of this paper are the two theorems stated here and the results of extensive computations which are presented in Section 2 and in Appendix A.
Let T = (V, E) be a tree with bipartition classes V 1 and V 2 and a bipartite labeling l : V → {0, . . ., |V | − 1}.Define the edge parity of T to be ( So if l is an α-labeling this is the sum of all edge labels modulo 2; it is 0 if |V | ≡ 0, 1 mod 4 and 1 if |V | ≡ 2, 3 mod 4. Define the vertex parity of T to be ( v∈V deg(v)l(v)) mod 2, or equivalently, to be the parity of the number of vertices of odd degree with odd label.
Theorem 1.1.In a tree T with α-deficit 0 the edge parity and the vertex parities are equal.
This theorem provides an explanation for the existence of trees with positive α-deficit, and moreover, as a consequence we can construct infinitely many such trees.The second theorem is very different to the results mentioned earlier in this section; it implies that under certain conditions, if there is a tree with n vertices and α-deficit d, then for all integers n > n there will also be a tree of the same deficit.
Theorem 1.2.For all k, d ∈ N and n ≥ k 2 + k, the number of trees T with n vertices, α def (T ) = d and maximum degree n − k is the same.
These theorems were suggested by computations of the α-deficit of all trees with up to 26 vertices and of trees with up to 36 vertices for several classes.These computations go far beyond anything that has been done before.To the best of our knowledge, until now only data on the α-deficit of trees up to 17 vertices was known; this appears in Van Bussel (2000).The results of our computations, presented in Appendix A, shed new light on the α-deficit; they support conjectures in the literature and also give rise to some surprising new conjectures about the α-deficit of general trees and trees with bounded degree.
The two theorems above are proved in Sections 4 and 5, a new proof of a theorem on m-comets is given in Section 3.1 and new conjectures are presented in the final section.

Computational results
Appendix A contains the numbers of trees with nonzero α-deficit for each vertex number up to 26.
The smallest trees for a given deficit d were stars with the central vertex of degree 3 • d and each edge subdivided.These trees were called comets by Rosa and Širáň (1995).Note that comets are not necessarily the trees that have the smallest maximum degree for a given deficit.
In the tables in Appendix A the following things catch the eye: • At the end of the lines with α-deficit one and two some series of numbers seem to grow that are repeated in every table for larger vertex numbers.
• While for small vertex numbers it looks like trees with maximum degree 3 and deficit are not unusual, they become sparse later.
• For maximum degree 3 and 4 no trees with α-deficit larger than one occur.
In order to better understand the first observation we investigated the trees with nonzero deficit and a given maximum degree depending on the number of vertices.For |V | ∈ {7, . . ., 26} there is always exactly one tree with maximum degree |V | − 4 and nonzero deficit.We tested the conjecture that this tree is in fact always the smallest tree in the series -that is the 3-comet depicted in Figure 1 -with the center of a star with |V | − 6 vertices identified with the (unique) vertex of degree 3.As far as the list goes this turned out to be the case.
In fact a similar statement is true for all k ∈ {4, . . ., 10}: for each of these k there is a smallest number m + (k) such that the number of trees with n ≥ m + (k) vertices, nonzero deficit and maximum degree n − k stays constant (within the reach of these tables).In fact we have m + (4) = 7, m + (5) = 9, m + (6) = 11, m + (7) = 13, m + (8) = 15, m + (9) = 18 and m + (10) = 19.In all cases the trees with m + (k), m + (k) + 1, . . ., 26 vertices were just the trees with m + (k) vertices plus the center of a star identified with a vertex of maximum degree of the original tree.Nevertheless there are examples when adding edges to a vertex of maximum degree in a tree can increase or decrease the deficit.In Section 5 we will prove that if you add enough pendant edges to the same vertex, the deficit will stay constant after some time.This gives an upper bound for the function m(k) defined there which is similar to m + (k) but without the restriction to numbers in the table and trees with positive deficit.
For k > 10 no conclusions based on the computed results are possible, but the step from m + (10) to m + (11) seems to be astonishingly large.
The last two observations were a motivation to run more tests on trees with the maximum degree restricted to 3 and 4. The results are given in Tables 1 and 2.
|V | 7 8 9 10 12 15 23 31 Tab. 1: The number of trees with maximum degree 3 and α-deficit 1.For vertex numbers n ≤ 36 that are not mentioned, no trees with deficit exist.Trees with maximum degree 3 and α-deficit larger than 1 do not exist within this range.
Note that although trees with maximum degree 3 and α-deficit seem to get sparse for |V | > 15, in steps of 8 vertices deficitary trees seem to exist.Most (though not all) of the trees on 37 vertices have been tested without finding a tree with deficit.
Figures 1, 2 and 3 give all deficitary trees with maximum degree 3 that were found in this computation.Note that from 15 vertices on all deficitary maximum degree 3 trees that were found in this search come from subdividing all edges of smaller trees.It would be interesting to know whether this is true also for |V | > 36.
The most striking property of Table 2 seems to be that for n ∈ {26, 28, 30, 32} no deficitary trees exist, while for odd vertex numbers (and smaller even vertex numbers) a lot of deficitary trees exist.A possible explanation is given in Section 4.
Inside the range of our computations maximum degree 5 is the smallest value that also allows deficit 2. Nevertheless up to 26 vertices only a small number of trees T with ∆(T ) = 5 have deficit 2 -at most 2 per vertex number.This motivated us to test trees with maximum degree 5 and more than 26 vertices.The results are given in Table 3.So it seems possible that α-deficit equal to two is only a local phenomenon for small trees with maximum degree 5 and that for more than 26 vertices the maximum deficit is also at most 1.All known trees T with ∆(T ) = 5 and α def (T ) = 2 are given in Figure 4.  3 The α-deficit of comets Rosa and Širáň (1995) defined an m-comet, C m , as a tree with 2m + 1 vertices: a central vertex w 0 and two sets of m vertices {w 1 , w 2 , . . ., w m } and {w m+1 , w m+2 , . . ., w 2m } such that E = {w 0 w i , w i w m+i ; 1 ≤ i ≤ m}.The first tree in Figure 1 is for example an m-comet for m = 3.In our lists the smallest trees with a given α-deficit α are unique -they are the (3 • α)-comets.Theorem 3.1, which is a reformulation of Corollary 9 of Rosa and Širáň (1995), determines the α-deficit of m-comets.
Theorem 3.1.Rosa and Širáň (1995) For every m ≥ 1 and comet C m we have α def (C m ) = m 3 .We give a new proof by applying the following theorem on nonattacking queens on a triangle.Theorem 3.2.Navisch and Lev (2005) For n ≥ 0, let q n be the maximum number of non-zero elements in a triangular (0, 1)-matrix of side length n with at most one non-zero element in each row, in each column and in each diagonal.Then q n = 2n+1 be the same, so by removing vertex labels of leaves, we can obtain a partial labeling with the same deficit, but all existing edge labels being unique.Let M = (a i,j ) 0≤i≤m,m+1≤j≤2m be a matrix and a i,j = |j − i|.The j-th column contains the possible labels of the two edges which are incident to w j .
We have to label the vertices of A. Let f (w 0 ) = k, then the k-th row contains all m pairwise different labels of the edges w 0 w j , (m + 1 ≤ j ≤ 2m).That is a k,j = |j − k|, for m + 1 ≤ j ≤ 2m, and let L be the set of these edge-labels.The remaining edges have to get induced labels with values in {0, 1, . . ., 2m} − L. Removing all entries a r,s ∈ L from M corresponds to removing diagonals in the matrix and we obtain an upper triangular matrix M u of side length m u = k and a lower one M l of side length m l = m − k (see Figure 5).
Marking an element a i,j in these triangular matrices if and only if the neighbour (different from w 0 ) of w j is labeled i, we get a set of marks with the property that on every diagonal, in every column and in every row we have at most one mark (because every edge label and vertex label occurs at most once here).
Therefore we can label at most m + 1 + q u + q l vertices, without creating a repetition of edge labels.We have that Observe that this upper bound on q u + q l can be reached for any value of m.Indeed, if m ≡ 0 mod 3 or m ≡ 1 mod 3, it is the case when k = 0 and if m ≡ 2 mod 3 when k = 1.
Note that this construction can also be reversed -that is: a solution to the nonattacking queens problem also gives a labeling -and thus the statement is proved.

A criterion for non-zero α-deficit
In this section we prove Theorem 1.1, which states that a tree with an α-labeling l must have the same edge and vertex parities modulo 2. This theorem can be considered to be a generalisation of the criterion in Theorem 3.3 in Huang et al. (1982).
Proof of Theorem 1.1: Since l is an α-labeling, the sum of the edge labels, modulo 2, gives the edge parity.The edge parity is therefore )) mod 2 where v i ∈ V i and e = {v 1 , v 2 }.This gives which is in fact the vertex parity.
Note that while the edge parity is a property of the tree that does not depend on the labeling, the vertex parity in general does.
Remark 4.1.If T is a tree with bipartition classes V 1 and V 2 so that the parity of the vertex degrees is the same for all vertices inside a bipartition class, then the vertex parity does not depend on the labeling.In this case we will speak about the vertex parity of the tree.
The proof is immediate because if all degrees are odd, the vertex parity is always the parity of the number of odd integers in {0, . . ., |V | − 1}.If one class contains vertices of even degree, w.l.o.g.V 1 , the vertex parity is either the parity of the number of odd integers in {0, . . ., We say that a tree T has the parity property if the parity of the vertex degrees is the same for all vertices inside a bipartition class and the vertex parity and edge parity differ.
Corollary 4.2.Trees that have the parity property 1. do not have an α-labeling.
2. have a bipartition class of even degrees.

have an odd number of vertices.
Proof: 1. is trivial 2. Assume that T does not contain a vertex of even degree.Then the vertex parity is As this is the edge parity, T does not have the parity property.
3. Since one of the bipartition classes, w.l.o.g V 1 , contains only vertices of even degree, the number of edges v∈V1 deg(v) is also even.Therefore the number of vertices, which is |E| + 1, is odd.
Figures 1, 2 and 3 show the deficitary trees with maximum degree 3 that have fewer than 37 vertices.In fact all but 5 trees are subdivisions of smaller trees with 4k vertices.The following explains why these trees have an α-deficit.
Corollary 4.3.A tree with only vertices of odd degree and degree 2 has the parity property if and only if T can be obtained from a tree S with 4k vertices, all of odd degree, by replacing each edge by a path of length 2.
Proof: Assume that T can be obtained by this construction.By construction the parities of the vertex degrees are the same in each of the bipartition classes.Since |V | = 8k − 1 the edge parity of T is 1.On the other hand a simple computation shows that the vertex parity is 0. So T has the parity property.
The fact that a tree with only vertices of odd degree and degree 2 that has the parity property is a subdivision of a smaller tree S with only odd degrees follows immediately because one bipartition class must have only vertices of degree 2 and the other only odd degrees.Suppose that the number v S of vertices of S is not a multiple of 4. Then v S = 4k − 2 for some k, and T has 8k − 5 vertices and 8k − 6 edges.Hence the edge parity is 1 and the vertex parity, ( 4k−3 i=0 i), is also 1.This is a contradiction.As a corollary we obtain the following result, already presented in Huang et al. (1982): Corollary 4.4.Subdivisions of trees with 4k vertices -all of odd degree -have positive α-deficit.
Note 4.5.The infinite series of subdivided trees shown in Figure 6 all satisfy the parity property.These trees have 8j − 1 vertices (j ≥ 1).If the vertex labelled 0 is removed then the remaining 8j − 2 edges have labels 1, 2, . . ., 8j − 3; with the vertex 0 included the label 4j + 1 appears twice.Hence there are infinitely many trees with maximum degree 3 and α-deficit 1.
In Figures 1, 2 and 3 it is easily seen that the known deficitary trees with maximum degree 3 and at least 15 vertices are exactly the trees with the parity property.For maximum degree 4 there are too many trees to present in detail or to check by hand.We checked the deficitary trees by computer and the result was that while in the beginning only a small ratio of the deficitary trees had the parity property, this ratio increased with the number of vertices and for more than 25 vertices the known deficitary trees with maximum degree 4 all have the parity property.For maximum degree 5 something similar happens from 26 vertices on: within the range of the tables the number of deficitary trees with an even number of vertices (which cannot have the parity property) seems to decrease and for odd vertex numbers the ratio of deficitary graphs with the parity property increases from 17% for 27 vertices to 57% for 31 vertices.So also for ∆(T ) = 5 the parity property seems to gain importance as the number of vertices increases.
We discovered the parity property as a consequence of the computational results and at that time did not know about Theorem 3.3 in Huang et al. (1982).The program did not use the parity property.Now the algorithm could be improved by using the parity property in the beginning and also as a bounding criterion in the recursion when all vertices inside a partition class that still need to be labeled have the same parity.
5 The α-deficit of trees with a vertex of large degree compared to the number of vertices The computational results in Section 2 and in Appendix A suggest that for every k ∈ N there exists some smallest m(k) so that for every n ≥ m(k) and every d ∈ N the number of trees T with n vertices, α def (T ) = d and maximum degree n − k is the same.In this section we will prove that such a value m(k) does exist for every k.
To this end we will first prove that after adding a certain number of pendant vertices to a fixed vertex in a tree the α-deficit does not change.
Lemma 5.1.Let T be a tree with bipartition classes V 1 and V 2 and a vertex v ∈ V 1 .Let T p denote the tree obtained from T by adding p pendant vertices to v.
For all p, p ≥ (|V Proof: We will prove that any bipartite labeling of T p implies a bipartite labeling of T p+1 with the same deficit and vice versa.Assume |V 1 | > 1. (Otherwise T p is a star and so has deficit 0.) We will denote the set of pendant vertices adjacent to v in T p by P , and in T p+1 by P +1 .The bipartition classes of T p and The Pigeonhole Principle implies that there are at least |V 1 | − 1 vertices in P with consecutive labels.Assume that u is the largest of these consecutive labels.For w ∈ T p+1 define With d(l +1 ) the number of different edge labels induced by l +1 and d(l) the number of different edge labels induced by l we will prove that d(l +1 ) = d(l) + 1, which implies that the deficits induced by the labelings are the same.
Define the sets of edge labels Note that S = S +1 and that Let M be the set of edge labels of pendant vertices with consecutive labels Now we will distinguish between two cases: In both cases d(l +1 ) = d(l) + 1.
Conversely, assume that a labeling l +1 of T p+1 is given.Then there is a sequence of at least |V 1 | vertices in P with consecutive labels.Let u be the largest label of these vertices and assume that the vertex labeled u is removed.Define It is easy to see that applying the previously discussed extension to this reduced labeling l produces l +1 , so the two labelings have in fact the same deficit.
We restate Theorem 1.2 here and give a proof.It enables us to present a function m(k) as described in the beginning of the section.
Theorem.For all k, d ∈ N, the number of trees T with n ≥ k 2 + k vertices, α def (T ) = d and maximum degree n − k is the same.This implies that m(k) ≤ k 2 + k.
Proof: Assume that k ≥ 3. Otherwise all trees are stars or stars with one edge subdivided and it can easily be seen that these have deficit 0.
As k ≥ 3 we have n − k ≥ n/2 + 1.It follows that there can be only one vertex v of maximum degree n − k in T .There are exactly k − 1 vertices different from v and not adjacent to v.This implies that at most k − 1 of the neighbours of v have another neighbour, so there are at least n − 2k + 1 pendant vertices.
Let M (n, k, d) denote the set of all trees with n vertices, α-deficit d and maximum degree n − k.We claim that adding a new pendant vertex to the unique vertex of maximum degree defines a bijection between M (n, k, d) and M (n+1, k, d).It is easy to see that the maximum degree and the number of vertices increase by one.What remains to be shown is that the α-deficit does not change.
Let T 0 be the tree obtained from T by removing n − 2k + 1 pendant vertices adjacent to v. Then T 0 has 2k − 1 vertices separated into bipartition classes of cardinality |V 1 | and 2k − 1 − |V 1 |.
By Lemma 5.1 the theorem is proved if the number of pendant vertices adjacent to v in T is at least Together with the tables and the numbers of trees with given maximum degree obtained by freetree (Li and Ruskey (1999)), Theorem 1.2 implies the following corollary.Note that the values m + (k) obtained from the tables were only for nonzero deficit and that it also has to be taken into account from when on the number of trees with n vertices and maximum degree n − k stays constant.

Conclusions, questions, conjectures
The computational results in this paper suggest the following conjectures: Conjecture 6.1.

Appendix A: Results
The number of trees with nonzero α-deficit for each vertex number up to 26 is given here.There are no trees with nonzero deficit and less than 7 vertices.On 7 vertices there is exactly one tree with α-deficit 1.It has maximum degree 3.
Trees with 8 vertices

Fig. 5 :
Fig. 5: The matrix M used in the proof of Theorem 3.1.The two triangular matrices Mu and M l are depicted for k = 2.
The number of trees with maximum degree 4 and α-deficit 1. Trees with maximum degree 4 and α-deficit larger than 1 do not exist within this range.
Tab. 3: The number of trees with maximum degree 5 and positive α-deficit.Trees with maximum degree 5 and α-deficit larger than 2 do not exist within this range.