The Birank Number of a Graph Michael J. Fisher∗

Nicholas Fisher∗

Michael Fraboni†

Darren A. Narayan‡

Abstract Given a graph G, a function f : V (G) → {1, 2, ..., k} is a kbiranking of G if f (u) = f (v) implies every u − v path contains vertices x and y such that f (x) > f (u) and f (y) < f (u). The birank number of a graph, denoted ϕr (G), is the minimum k such that G has a k-biranking. In this paper we determine the birank numbers for various families of graphs.

1

Introduction

Given a graph G, a function f : V (G) → {1, 2, ..., k} is a k-biranking of G if f (u) = f (v) implies every u − v path contains vertices x and y such that f (x) > f (u) and f (y) < f (u). The birank number of a graph, denoted ϕr (G), is the minimum k such that G has a k-biranking. The idea of k-birankings stems from the well-studied area of k-rankings. A function f : V (G) → {1, 2, ..., k} is a k-ranking of G if f (u) = f (v) implies every u − v path contains a vertex w such that f (w) > f (u). The rank number of a graph, denoted χr (G), is the minimum k such that G has a minimal k-ranking. Ghoshal, Laskar, and Pillone [5] introduced minimal k-rankings. A kranking was defined to be (locally) minimal if the reduction of any label greater than 1 violates the described ranking property. Another definition of a minimality is that a k-ranking f is globally minimal if for all v ∈ V (G), f (v) ≤ g(v) for all rankings g. It was shown by Jamison [10] and Isaak, Jamison, and Narayan [9] that these two definitions of minimal rankings are ∗ Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, Email: [email protected] & [email protected] † Department of Mathematics, Moravian College, Bethlehem, PA 18018, Email: [email protected] ‡ School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, Email: [email protected].

1

equivalent. The arank number of a graph, denoted ψr (G), is the maximum k such that G has a minimal k-ranking. Rank numbers are known for several families of graphs including paths, cycles, split graphs, complete multipartite graphs, M¨obius graphs, powers of paths and cycles, and some grid graphs [2],[3],[5],[14],[1], and [15]. However the arank number is only known for a few families of graphs: complete multipartite graphs, split graphs, rook’s graphs (the Cartesian product of Kn and Kn ), paths, and within 1 for cycles [5], [13], [12], and [11]. We note that the arank number of a path was determined by Williams  Kostyuk, Narayan, and

 who showed ψr (Pn ) = blog2 (n + 1)c + log2 n + 1 − 2blog2 nc−1 [12]. In 2003, Jamison [10] introduced birankings of graphs. Local and global minimality are defined analogously for birankings. Isaak, Jacob, Jamison, and Narayan [8] showed that the two definitions are equivalent. Jamison noted that the arank analog for birankings is uninteresting, being equal to number of vertices in the largest component of G. However of the three other numbers χr (G), ψr (G), and ϕr (G), only the first two have been studied. This is the first paper to investigate birank numbers of a graph. We determine birank numbers for paths, cycles, and crowns. This includes a surprising result, ϕr (Pn ) = ψr (Pn ).

2

Preliminaries

We begin with an elementary result that will be used throughout the paper. Lemma 1 If H is a subgraph of G then ϕr (H) ≤ ϕr (G). Proof. Assume ϕr (G) = k. Let f be a k-biranking of G. Note that f|H is a k-biranking of H. Given u, v ∈ H with f (u) = f (v), if there is a u − v path in H then that same path exists in G, so there exists w1 and w2 on the path with f (w1 ) < f (u) < f (w2 ). But since w1 ,w2 are in our u − v path, they are in H. Hence ϕr (H) ≤ k. Next we give an elementary result that follows directly from the definition of a biranking of a graph. Proposition 1 Let G be a graph with diam(G) ≤ 2. Then ϕr (G) = n. As a consequence we note that if G is a complete multipartite graph then ϕr (G) = n. It is not too difficult to establish a similar result for crowns. Recall that a crown graph is Kn,n minus a perfect matching. Let CRn be the crown graph on vertices {x1 , ..., xn , y1 , ...., yn } with xi adjacent to yj if and only if i 6= j.

Theorem 1 ϕr (CRn ) = 2n − 1. Proof. We construct a biranking f : CRn → {1, . . . , 2n − 1}. For i = 1, . . . , n let f (xi ) = i and for i = 1, . . . , n − 1 let f (yi ) = n + i. Finally, let f (yn ) = n. Note that any path from xn to yn must contain some xi and some yj for i, j ∈ {1, . . . , n − 1}. But f (xi ) < f (xn ) = f (yn ) < f (yj ) so we have a biranking. We establish the lower bound using a proof by contradiction. Assume there is a biranking f : CRn → {1, . . . , 2n−2}. Note that f (xi ) 6= f (xj ) for i 6= j since any xi and xj are distance two apart. Similarly, f (yi ) 6= f (yj ) for i 6= j. Also, f (xi ) 6= f (yj ) for i 6= j since these vertices are a distance of one apart. Therefore, there exist s, t ∈ {1, . . . , n} such that f (xs ) = f (ys ) and f (xt ) = f (yt ). Without loss of generality we may assume that f (xs ) < f (xt ). There exists a k such that either f (xk ) or f (yk ) is less than f (xs ). Without loss of generality suppose that f (xk ) < f (xs ). Consider the path yt − xk − ys − xt . This path connects xt and yt but contains no vertex with rank more than f (xt ) = f (yt ) so f cannot be a biranking and hence a contradiction.

3

The birank number of a path

We consider the birank number of a path on n vertices Pn . We will use Si to denote the set of vertices labeled i. Lemma 2 Let f be a minimal k-biranking of Pn . Then  i−1   when i ≤ k2  2 |Si | ≤   .  k−i 2 when i > k2 Proof. It follows immediately from the biranking property that |Sj | ≤ Xj−1 Xk |Si | and |Sj | ≤ |Si |. It is clear that in a k-biranking |S1 | = i=1

i=j+1

|Sk | = 1. The result then follows. Our strategy is to give an upper bound by constructing an appropriate k-biranking and then show no biranking with less than k labels exists. The combination of the next two theorems gives the birank number for paths. Theorem 2 ϕr (P2k−1 +2k−2 −2 ) = 2k − 3. Proof. Let n = 2k−1 + 2k−2 − 2. We first construct a minimal (2k − 3)biranking of Pn with vertices v1 , v2 , ..., vn by labeling the vertices as follows:

f (v3j−2 ) = k − 1 for all j = 1, ..., n+2 3 . f (v3j−1 ) = α + k where α is the largest integer such that 2α divides j, for all j = 1, ..., n−1 3 . f (v3j ) = 2k − 2 − f (v3j−1 ) for all j = 1, ..., n−1 3 . Now we seek to show that a minimal (2k − 4)-biranking of P2k−1 +2k−2 −2 does not exist. Assume there exists a minimal (2k − 4)-biranking of P2k−4 Pk−2 P2k−1 +2k−2 −2 . By Lemma 2 we have i=1 |Si | ≤ 2 i=1 2i−1 = 2k−1 − 12 < 2k−1 + 2k−2 − 2. This is a contradiction. Theorem 3 ϕr (P2k −2 ) = 2k − 2. Proof. Let n = 2k − 2. We first construct a minimal (2k − 2)-biranking of Pn with vertices v1 , v2 , ..., vn by labeling the vertices as follows: f (v4j−3 ) = k for all j = 1, ..., n+2 4 . f (v4j−2 ) = k − 1 for all j = 1, ..., n+2 4 . f (v4j−1 ) = α + 1 + k where α is the largest integer such that 2α divides j, for all j = 1, ..., n−2 4 . f (v4j ) = 2k − 1 − f (v4j−1 ) for all j = 1, ..., n−2 4 . Now we seek to show that a minimal (2k − 3)-biranking of P2k−1 −2 does not exist. Assume there exists a minimal (2k − P  3)-biranking of P2k−1 −2 . By P2k−3 k−2 i−1 Lemma 2 we have i=1 |Si | ≤ 2 + 2k−2 = 2k−1 + 2k−2 − 2 < i=1 2 2k − 2. This is a contradiction. We note that these two theorems correspond exactly with Theorem 13 from [12] that gives the arank number of a path. Combining these theorems with the monotonicity property from Lemma 1 gives the following result. Theorem 4 j   k ϕr (Pn ) = ψ (Pn ) = blog2 (n + 1)c + log2 n + 1 − 2blog2 nc−1 .

3.1

A connection to Huffman codes

Fisher, Fisher, and Fraboni discovered an interesting connection between the biranking number of a path on n and Huffman prefix-free codes [4]. This connection was realized using Neil Sloane’s website, The On-line Encyclopedia of Integer Sequences (OEIS) [16]. In an effort to describe this connection, first recall that a binary code is an assignment of symbols to a set of bitstrings. A prefix-free code is a binary code with the property that no codeword is an initial substring of any other codeword.

A binary tree can be used to construct a prefix-free code in the following manner. Create any binary tree where the leaves correspond bijectively to the given symbols. Label any edge leading to a left-child with a 0 and label any edge leading to a right-child with a 1. Then a symbol’s codeword is the bitstring that corresponds to the unique path from the root to the symbol’s vertex. It is fairly easy to see that the set of codewords produced in this manner is a prefix-free code. The following binary code is an example of a prefix-free code on 5 symbols. Symbol a b c d e Codeword 0000 0001 001 01 1 In most applications, each initial symbol appears with a given frequency (the frequencies need not be distinct, but they must add to one). Then, one measure of the “efficiency” of a prefix-free code is the average weighted length of a codeword, where the length of each codeword is multiplied by its corresponding frequency. Consider the above code where the symbols have frequency 0.1, 0.05, 0.2, 0.3, and 0.35, respectively. The average weighted length of the above code is 0.1 × 4 + 0.05 × 4 + 0.2 × 3 + 0.3 × 2 + 0.35 × 1 = 2.15. In 1952, David A. Huffman developed an algorithm that constructs a prefix-free code whose codewords have the smallest possible average weighted length [7]. His algorithm makes use of a frequency-sorted binary tree and is very clearly described in [6]. According to the OEIS (A126236), the conjectured maximum length of a codeword in a Huffman prefix-free code on n symbols, where the k th symbol has frequency k, is    2(n + 1) an = blog2 (n + 1)c + log2 . 3 So, for example, consider the Huffman prefix-free code Symbol Frequency Codeword

a b 1/36 2/36 00000 00001

c 3/36 0001

d 4/36 100

e 5/36 101

f 6/36 001

g 7/36 11

h 8/36 01

Note that the maximum length of any codeword is 5 = a8 . It is also conjectured (see OEIS A126235) that    2(n + 1) bn = log2 3 is the minimum length of a codeword in a Huffman prefix-free code on n symbols, where the k th symbol has frequency k. For the above example note that 2 = b8 .

Since blog2 Theorem 4.

4



2(n+1) 3



 j k c = log2 2(n+1) , the above result connects to 3

Cycles

In this section we will establish the following result for cycles. Theorem 5 For any cycle Cn , ϕr (Cn ) = ϕr (Pn−2 ) + 2. As before, we use Si to denote the set of vertices labeled i. Lemma 3 Let f be a biranking of   1 2i−2 |Si | ≤  k−i−1 2

Cn . Then when when when

i = 1 or k  2 ≤ i ≤ k2 . i > k2

Proof. By the biranking property, |Sj | ≤

Xj−1 i=1

|Si | and |Sj | ≤

Xk i=j+1

It is clear that in a k-biranking |S1 | = |S2 | = |Sk−1 | = |Sk | = 1. The result then follows. Lemma 4 For a k-biranking of Cn , we have ( k if k is even 22 . n≤ k 3 · 2b 2 c−1 if k is odd Proof. By Lemma 3 any rank r with d k2 e + 1 = k − b k2 c + 1 ≤ r < k appears at most 2k−r−1 times. So the number of vertices we may birank in a path using the ranks 1, . . . , b k2 c and k − b k2 c + 1, . . . , k is bounded by k

2+2

b2c X

k

2i−2 = 2b 2 c .

i=2

If k is even then we have our result. If k is odd then we need to account k for the ranking d k2 e. Since the ranking b k2 c appears at most 2b 2 c−2 times k we see the rank d k2 e may appear at most 2b 2 c−1 times. Now we observe k k k 2b 2 c−1 + 2b 2 c = 3 · 2b 2 c−1

and we have our result for k odd. Now we use this result to establish a lower bound on the birank number of a cycle.

|Si |.

Lemma 5 ϕr (Cn ) ≥ ϕr (Pn−2 ) + 2. Proof. Given n ∈ N, let k = ϕr (Cn ). We have two cases. First, if k is k even then by Lemma 4, n ≤ 2 2 so k

n − 2 ≤ 22 − 2 = 2

k−2 2 +1

−2

and there is a (k − 2)-biranking of Pn−2 . In other words, k − 2 ≥ ϕr (Pn ) and we have our result when k is even. Second, if k is odd then by Lemma k 4, n ≤ 3 · 2b 2 c−1 so k

n − 2 ≤ 3 · 2b 2 c−1 − 2 = 3 · 2b

k−2 2 +1c−1

− 2 = 3 · 2b

k−2 2 c

−2

and there is a k − 2 biranking on Pn−2 . In other words, k − 2 ≥ ϕr (Pn ) and we have our result. Lemma 6 ϕr (Cn ) ≤ ϕr (Pn−2 ) + 2. Proof. Given a cycle Cn , let k = ϕr (Pn−2 ) + 2. We construct a biranking f : Cn → {1, . . . , k}. Fix u, v adjacent vertices in Cn , define f (u) = 1, and f (v) = k. The graph Cn − {u, v} is a path with n − 2 vertices, so there is a biranking g : Cn − {u, v} → {1, . . . , k − 2}. For x ∈ Cn − {u, v}, define f (x) = g(x) + 1. Clearly f is a biranking of Cn . Given v1 , v2 ∈ Cn with f (v1 ) = f (v2 ) note that by definition of f , vi 6∈ {u, v} for i = 1, 2. There are exactly two paths connecting v1 and v2 . One of these contains u, v, and so satisfies the definition of a biranking. The other path does not contain u, v, so lies entirely in Cn − {u, v} and must satisfy the definition of biranking since g was a biranking.

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[5] J. Ghoshal, R. Laskar, and D. Pillone. Minimal rankings, Networks, Vol. 28, (1996), 45-53. [6] J. L. Gross and J. Yellen, Graph Theory and Its Applications, 2nd ed., Chapman & Hall / CRC, Boca Raton, FL (2006). [7] D. A. Huffman, A method for the construction of minimum-redundancy codes, Proceedings of the I.R.E., September 1952, 1098-1101, [8] G. Isaak, J. Jacob, R. Jamison, and D. A. Narayan, Minimal birankings of graphs, in preparation. [9] G. Isaak, R. Jamison, and D. A. Narayan, Greedy rankings and arank numbers, Information Processing Letters, 109 (2009), 825-827. [10] R. E. Jamison, Coloring parameters associated with the rankings of graphs, Congressus Numerantium. 164 (2003), 111–127. [11] V. Kostyuk and D. A. Narayan, Minimal k-rankings for cycles, Ars Combinatoria, in press. [12] V. Kostyuk, D. A. Narayan, and V. A. Williams, Minimal rankings and the arank number of a path, Discrete Math. 306, (2006), 1991-1996. [13] R. Laskar, D. Pillone, G. Eyabi, and J. Jacob, Minimal Rankings of Rook’s Graphs, submitted for publication. [14] S. Novotny, J. Ortiz, and D. A. Narayan, Minimal k-rankings and the rank number of Pn2 , Information Processing Letters, 109 (No.3) (2009), 193-198. [15] J. Ortiz, H. King, A. Zemke, and D. A. Narayan, The rank number of a prism graph, submitted for publication. [16] N. Sloane, The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/˜njas/sequences/.