Maximizing the number of edges in optimal $k$-rankings

TL;DR

This paper investigates the maximum number of edges that can be added to specific graphs without increasing their rank number, providing explicit characterizations of edge effects on the rank number.

Contribution

It determines the maximum edges that can be added to certain graphs without changing their rank number and characterizes which edges influence the rank number.

Findings

Maximum edges added to paths, cycles, and bipartite graphs without changing rank number

Explicit characterization of edges that alter the rank number

Results applicable to specific graph classes such as unions of cliques

Abstract

A $k$-ranking is a vertex $k$-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest $k$ such that $G$ has a $k$-ranking. For certain graphs $G$ we consider the maximum number of edges that may be added to $G$ without changing the rank number. Here we investigate the problem for $G=P_{2^{k-1}}$, $C_{2^{k}}$, $K_{m_{1},m_{2},\dots,m_{t}}$, and the union of two copies of $K_{n}$ joined by a single edge. In addition to determining the maximum number of edges that may be added to $G$ without changing the rank number we provide an explicit characterization of which edges change the rank number when added to $G$, and which edges do not.

Equations

$$g(\alpha_{i})=\left\{\begin{array}[c]{ll}0&\mbox{if $\alpha_{i}=1$}\\ 1&\mbox{if $\alpha_{i}=0$}.\end{array}\right.$$

$$E(v)=\{vw\mid w\in V(\mathcal{C})\text{ and w not adjacent to v}\}.$$

$$E_{j}=\bigcup_{\begin{tabular}[]{c}$v\in\mathcal{C}$, $f(v)=j-1$\\ $\mathcal{C}\in\mathcal{C}(A_{j})$\\ \end{tabular}}E(v).$$

$$\chi \left(P_{2^n-1}\cup \bigcup _{j=4}^t{E}_j\right)=\chi (P_{2^n-1}).$$

$$G_0=P_{2^n-1}\cup \bigcup _{j=4}^k{E}_j\text{ and }{G}_1=P_{2^n-1}\cup \bigcup _{j=4}^{k+1}{E}_j.$$

$$a_n=2\mu (P_{2^{n-1}-1})+\mid E(v_{2^{n-1}})\mid .$$

$$H_C=H_P\cup \{e\mid e\notin C_{2^k}\text{ and vertices }\{v_{2^k},v_i\},\text{ with }i\in \{2,\dots ,2^k-2\}\}.$$

$$f(x)=\left\{\begin{array}[c]{cc}1&\text{ if }x\in W\\ i&\text{ if }x=v_{i}\text{ for some }v_{i}\in V.\end{array}\right.$$

$$\mu (K_{m_1,m_2,\dots ,m_t})=\sum _{s=2}^t\frac{(m_s-1)m_s}{2}.$$

$$f(x)=\left\{\begin{array}[c]{ll}i&\text{ if }x=w_{i}\text{ for some }w_{i}\in W\\ i&\text{ if }x=v_{i}\text{ for some }v_{i}\in V\setminus\{v_{n}\}\\ n+1&\text{ if }x=v_{n}.\end{array}\right.$$

$$H_1=\{e\mid e\notin G_n\text{ is an edge with vertices }{w}_n,v_i\text{ for some }i\in \{1,2,\dots ,n-1\}\}$$

$$H_2=\{e\mid e\notin G_n\text{ is an edge with vertices }{v}_n,w_i\text{ for some }i\in \{1,2,\dots ,n-1\}\}.$$

Full text

1 Introduction

A vertex coloring of a graph is a labeling of the vertices so that no two adjacent vertices receive the same label. A $k$-ranking of a graph is a coloring of the vertex set with $k$ positive integers such that on every path connecting two vertices of the same color there is a vertex of larger color. The rank number of a graph is defined to be the smallest $k$ such that $G$ has a $k$-ranking.

Early studies involving the rank number of a graph were sparked by its numerous applications including designs for very large scale integration layout (VLSI), Cholesky factorizations of matrices, and the scheduling of manufacturing systems [5, 7, 8]. Bodlaender et al. proved that given a bipartite graph $G$ and a positive integer $n$, deciding whether a rank number of $G$ is less than $n$ is NP-complete [2]. The rank number of paths, cycles, split graphs, complete multipartite graphs, powers of paths and cycles, and some grid graphs are well known [1, 2, 3, 4, 6, 9, 10].

In this paper we investigate an extremal property of rankings that has not yet been explored. We consider the maximum number of edges that may be added to $G$ without changing the rank number. Since the maximum number of edges that can be added to a graph without changing the rank number varies with each particular ranking, we will focus on families where an optimal ranking has a specific structure. Here we investigate the problem for $G=P_{2^{k-1}}$, $C_{2^{k}}$, $K_{m_{1},m_{2},\dots,m_{t}}$, and the union of two copies of $K_{n}$ joined by a single edge.

In addition to determining the maximum number of edges that may be added to $G$ without changing the rank number we provide an explicit characterization of which edges change the rank number when added to $G$, and which edges do not. That is, given a vertex $v_{n}$ in $n$th position in the graph, we provide an algorithm to add a new edge with $v_{n}$ as one of its vertices to the graph $G$ without changing its ranking. For this construction we use the binary representation of $n$ to determine the position of the second vertex of the new edge. We also construct the maximum number of edges, so called good edges, that can be added to the graph without changing its ranking. We enumerate the maximum number of good edges.

2 Preliminaries

In this section we review elementary properties and known results about rankings.

A labeling $f:V(G)\rightarrow\{1,2,\dots,k\}$ is a $k$-ranking of a graph $G$ if and only if $f(u)=f(v)$ implies that every $u-v$ path contains a vertex $w$ such that $f(w)>f(u)$. Following along the lines of the chromatic number, the rank number of a graph $\chi_{r}(G)$ is defined to be the smallest $k$ such that $G$ has a $k$-ranking. If $H$ is a subgraph of $G$, then $\chi_{r}(H)\leq\chi_{r}(G)$ (see [6]).

We use $P_{n}$ to represent the path with $n$ vertices. It is well known that a ranking of $P_{n}$ with $V\left(P_{n}\right)=\left\{v_{1},v_{2},...,v_{n}\right\}$ and $\chi_{r}(P_{n})$ labels can be constructed by labeling $v_{i}$ with $\alpha+1$ where $2^{\alpha}$ is the largest power of $2$ that divides $i$  [2]. We will call this ranking the standard ranking of a path.

We use $C_{2^{k}}$ to denote a cycle with $2^{k}$ vertices. A multipartite graph with $t$ components is denoted by $K_{m_{1},m_{2},\dots,m_{t}}$ where the $i$th component has $m_{i}$ vertices. The complete graph with $n$ vertices is denoted by $K_{n}$.

Let $\Gamma$ and $H$ be graphs with $V(H)\subseteq V(\Gamma)$ and $E(H)\cap E(\Gamma)=\emptyset$. We say that an edge $e\in H$ is good for $\Gamma$ if $\chi_{r}\left(\Gamma\cup\{e\}\right)=\chi_{r}(\Gamma)$, and $e$ is forbidden for $\Gamma$ if $\chi_{r}(\Gamma\cup\{e\})>\chi_{r}(\Gamma)$. We use $\mu(G)$ to represent the cardinality of the maximum set of good edges for $G$.

For example, in Figure 1 we show a ranking of a graph $P_{2^{4}-1}\cup H_{P}$ where $H_{P}$ is the set of all good edges for $P_{2^{4}-1}$. The set of vertices of $P_{2^{4}-1}$ is $\{v_{1},\dots,v_{15}\}$. We can see that $\chi_{r}(P_{2^{4}-1}\cup H_{P})=\chi_{r}(P_{2^{4}-1})=4$ and that $E(H_{P})$ is comprised of 20 good edges. That is, $\mu(P_{2^{4}-1})=20$. Theorems 4 and 6 give necessary and sufficient conditions to determine whether graphs $G=P_{2^{k}-1}\cup H_{P}$ and $P_{2^{k}-1}$ have the same rank number.

Figure 2 Part (a) shows the graph $G=C_{2^{4}}\cup H$ where $H$ is the set of all good edges for $C_{2^{4}}$. We can see that $\chi_{r}(C_{2^{4}}\cup H)=\chi_{r}(C_{2^{4}})=5$ and that $E(H)$ is comprised of 33 good edges. That is, $\mu(C_{2^{4}})=33$. Theorem 8 gives necessary and sufficient conditions to determine whether the graphs $G=C_{2^{k}}\cup H$ and $C_{2^{k}}$ have the same rank number.

Lemma 1** ([2, 3])**

If $k\geq 1$, then

$P_{2^{k}-1}$* has a unique *$k-$ranking and $\chi_{r}(P_{2^{k}-1})=k$. 2. 2.

$C_{2^{k}}$* has a unique *$k-$ranking and $\chi_{r}(C_{2^{k}})=k+1$.

3 Enumeration of the Set of Good Edges for $P_{2^{k}-1}$

In this section we give two ways to find the maximum set of edges that may be added to $G$ without changing the rank number. We give an algorithm to construct a good edge for $G$. The algorithm is based on the binary representation of $n$, the position of the vertex $v_{n}$. That is, given a vertex $v_{n}\in G$ in $n$th position, the algorithm add a new edge, with $v_{n}$ as one of its vertices, to the graph $G$ without changing its ranking. We show that if the graph $G$ is the union of $P_{2^{t}-1}$ and one edge of the form as indicated in Procedure 1, then the ranking of $G$ is equal to the ranking of $P_{2^{t}-1}$. This guarantees that the edges constructed using Procedure 1, are good edges. We also give sufficient and necessary conditions to determine whether a set of edges $H$ is a set of good set for $P_{2^{t}-1}$.

Since one of our aims is to enumerate the maximum number of edges that can be added to a graph without changing its rank, we give a recursive construction of the maximum set of “good edges”. The recursive construction gives us a way to count the the number of edges in the set of good edges.

We recall that $(\alpha_{r}\alpha_{r-1}\ldots\alpha_{1}\alpha_{0})_{2}$ with $\alpha_{h}=0\text{ or }1$ for $0\leq h\leq r$ is the binary representation of a positive integer $b$ if $b=\alpha_{r}2^{r}+\alpha_{r-1}2^{r-1}+\ldots+\alpha_{1}2^{1}+\alpha_{0}2^{0}$. We define

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Procedure 1. Let $V(P_{2^{k}-1})=\{v_{1},v_{2},\ldots,v_{2^{k}-1}\}$ be the set of vertices of $P_{2^{k}-1}$ and let $H_{P}$ be a graph with $V(H_{P})=V(P_{2^{k}-1})$. Suppose that $m<n+1$, $t=\lfloor\log_{2}{m}\rfloor$ and $m=(\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}\alpha_{0})_{2}$. If $\alpha_{j}$ is the nonzero rightmost entry of $m$, then an edge $e=\{v_{m},v_{n}\}$ is in $H_{P}$ if satisfies any of the following three conditions:

if $m$ is odd then either $n=2^{w}$ for $w>t$ or $n=\Omega(s)$ with $\Omega(s)=m+1+\sum_{i=1}^{s}g(\alpha_{i})2^{i}$ for $s=1,2,\ldots,t-1$ where $m=(\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}\alpha_{0})_{2}$, 2. 2.

$m=2^{j}\cdot(2l+1)$ and $2^{j}\cdot(2l+1)+2\leq n<2^{j}\cdot(2l+2)$, for $l>0$, 3. 3.

$m=2^{j}\cdot(2l+1)$ and $n=2^{w}$ for $2^{w}\geq 2^{j}\cdot(2l+2)$.

Procedure 2. Let $V(C_{2^{k}})=\{v_{1},v_{2},\ldots,v_{2^{k}}\}$ be the set of vertices of $C_{2^{k}}$. Let $H$ be a graph with $V(H)=V(C_{2^{k}})$. Suppose that $m<n+1$, $t=\lfloor\log_{2}{m}\rfloor$ and $m=(\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}\alpha_{0})_{2}$. If $\alpha_{j}$ is the nonzero rightmost entry of $m$, then an edge $e=\{v_{m},v_{n}\}$ is in $H$ if satisfies any of the following four conditions:

if $m$ is odd then either $n=2^{w}$ for $w>t$ or $n=\Omega(s)$ with $\Omega(s)=m+1+\sum_{i=1}^{s}g(\alpha_{i})2^{i}$ for $s=1,2,\ldots,t-1$, 2. 2.

$m=2^{j}\cdot(2l+1)$ and $2^{j}\cdot(2l+1)+2\leq n<2^{j}\cdot(2l+2)$, 3. 3.

$m=2^{j}\cdot(2l+1)$ and $n=2^{w}$ for $2^{w}\geq 2^{j}\cdot(2l+2)$ where $l\geq 0$, 4. 4.

$1<m<2^{k}-1$ and $n=2^{k}$.

Lemma 2

Suppose that $f$ is the k-ranking of $P_{2^{k}-1}$, and $m=(\alpha_{r}\alpha_{r-1}\ldots\alpha_{1}\alpha_{0})_{2}$. Let $t=\lfloor\log_{2}{m}\rfloor$.

If $\alpha_{i}=0$ for $i<j$ and $\alpha_{j}=1$ , then $f(v_{m})=j$. 2. 2.

$f(v_{j})<f(v_{\Omega(i)})$* for $m<j<\Omega(i)$ for $i=1,2,\ldots,t-1$.*

Proposition 3

Let $e\not\in P_{2^{t}-1}$ be an edge with vertices $v_{m}$ and $v_{n}$ where $m<n$. If $e$ is good for $P_{2^{t}-1}$ then $e\in H_{P}$.

Proof. We proceed with a proof by contradiction assuming that $e\not\in H_{P}$. Hence we have one of the following cases:

$m$ is odd and $2^{t+1}<n\not=2^{w}$ where $t=\lfloor\log_{2}{m}\rfloor$, 2. 2.

$m$ is odd, $2^{t}<m+1<n<2^{t+1}$ and $n\not=m+1+\sum_{i=1}^{s}g(\alpha_{i})2^{i}$ for $s=1,2,\ldots,t-1$ with $t=\lfloor\log_{2}{m}\rfloor$ and $m=(\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}\alpha_{0})_{2}$, 3. 3.

$m=2^{j}\cdot(2l+1)$ and $2^{t}<n<2^{t+1}$ where $2^{t}\geq 2^{j}\cdot(2l+2)$ with $l\geq 0$.

If Case 1 holds, then $e$ connects vertices $v_{m}$ and $v_{n}$ with $n>2^{t}$. Suppose that $2^{w}<n<2^{w+1}$. The standard ranking $f$ of a path implies that $f(v_{\beta})\in\{1,2,\ldots,w\}$ if $\beta<2^{w}$ and that $f(v_{\delta})\in\{1,2,\ldots,w\}$ for any $2^{w}<\delta<2^{w+1}$. Therefore, there are two vertices $v_{\gamma}$ and $v_{\rho}$ such that $f(v_{\gamma})=f(v_{\rho})=w$ with $\gamma<2^{w}<\rho<2^{w+1}$. The path containing the edge $e$ and vertices $v_{\gamma}$, $v_{m}$, $v_{n}$, and $v_{\rho}$ has two equal labels with no larger label in between, which contradicts the ranking property. Hence $m$ is odd and $n=2^{w}$ for $w>t$, and $e$ is good for $P_{2^{k}-1}$.

If Case 2 holds, then $e$ connects vertices $v_{m}$ and $v_{n}$ with $n<2^{t-1}$. If $m=2^{t+1}-1$, the argument is similar to the above case, so we suppose that $m\not=2^{t+1}-1$. If $n$ is odd then $f(v_{m})=f(v_{n})=1$, which is a contradiction.

For the remaining part of this case, we suppose that $m+2<n<2^{t+1}-1$ is even. This implies that $m\not=2^{t+1}-1$ and $m\not=2^{t+1}-3$ (note that the Proposition 3 is now proved for $n<8$). Therefore, there is at least one nonzero element in $A=\{\alpha_{2},\ldots,\alpha_{t-1},\alpha_{t}\}$ and let $i$ be the smallest subscript such that $\alpha_{i}\in A$ and $\alpha_{i}=0$. This give rise to two subcases for the location of n:

1. (a)

$n<\omega$ with $\omega=\left(m+1+g(\alpha_{i})2^{i}\right)$. 2. (b)

$\Omega(r)<n<2^{t+1}$ where $r$ is the largest number for which the inequality holds.

If subcase (a) holds, then $\alpha_{j}=1$ for $j<i$. This implies that first number equal to one in the binary notation of $m+1$ is in position $i+1$. This and Lemma 2 imply that $f(v_{m+1})=i+1$. Therefore, $f(v_{\omega})=i+2$, since $m+1<n<\omega$. The definition of the ranking function $f$, implies that $f(v_{\beta})\in\{1,2,\ldots,i\}$ if $m+1<\beta<\omega$ and that $f(v_{\delta})\in\{1,2,\ldots,i\}$ for any $\delta<m+1$. Therefore, there are two vertices $v_{\gamma}$ and $v_{\rho}$ such that $f(v_{\gamma})=f(v_{\rho})=i$ with $\gamma<m+1<\rho<\omega$. The path containing the edge $e$ and vertices $v_{\gamma}$, $v_{m}$, $v_{n}$, and $v_{\rho}$ has two equal labels with no bigger label in between, which is a contradiction.

Suppose that subcase (b) holds. From Lemma 2 we know that $f(v_{d})<f(v_{\Omega(r)})$ for $m<d<r$, in particular we deduce that $f(v_{\Omega(i)})<f(v_{\Omega(r)})$ if $i<r$. Let $w=f(v_{\Omega(r)})$. This, $P_{2^{k}-1}$ and the definition of $f$ imply that $f(v_{\beta})\in\{1,2,\ldots,w-1\}$ for any $\beta<\Omega(r)$ and that $f(v_{\delta})\in\{1,2,\ldots,w-1\}$ for any $\Omega(r)<\delta<\Omega(r+1)$. This implies that there are two vertices $v_{\gamma}$ and $v_{\rho}$ such that $f(v_{\gamma})=f(v_{\rho})=w-1$ with $\gamma<\Omega(r)<\rho<\Omega(r+1)$. The path containing the edge $e$ and vertices $v_{\gamma}$, $v_{m}$, $v_{n}$, and $v_{\rho}$ has two equal labels with no larger label in between, which is a contradiction.

Finally suppose that Case 3 holds. That is, we suppose that every edge $e$ connecting the vertex $v_{m}$ and $v_{n}$ is good, with $m=2^{j}\cdot(2l+1)$ and $2^{t}<n<2^{t+1}$ where $2^{t}\geq 2^{j}\cdot(2l+2)$ with $l\geq 0$. This implies that for any $s\leq 2^{t}$, the label $f(v_{s})\in\{1,2,\ldots,t\}$, in particular $f(v_{m})=j+1<t$. Since $2^{t}<n<2^{t+1}$, the coloring $f(v_{n})\in\{1,2,\ldots,t\}$. Then there are vertices $v_{s}$ and $v_{s^{\prime}}$ with $f(v_{s})=f(v_{s^{\prime}})=t$ for $s<2^{t}$ and $2^{t}<s^{\prime}<2^{t+1}$. This is a contradiction since the path containing the edge $e$ and connecting vertices $v_{s}$, $v_{m}$, $v_{n}$ and $v_{s^{\prime}}$, does not have a label larger than $t$.

Theorem 4

Let $G=P_{2^{k}-1}\cup H_{P}$. The set $E(H_{P})$ is the set of good edges for $P_{2^{k}-1}$.

Theorem 4 can be proved using Proposition 3, so we omit the proof. This Theorem is equivalent to Theorem 6 Part 1. The proof of Theorem 6 counts the maximum number of good edges. We now give some definitions that are going to be used in Lemma 5 and Theorem 6. Let $G$ be a graph with $f$ as its $k$-ranking. We define $A_{j}=\{v\in V(G)\mid f(v)\geq j\}$ and use $\mathcal{C}(A_{j})$ to denote the set of all component of $G\setminus A_{j}$. If $\mathcal{C}\in\mathcal{C}(A_{j})$ and $v\in V(\mathcal{C})$, then

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We denote by $E_{j}$ the union of all sets of the form $E(v)$ where $f(v)=j-1$, the vertex with maximum label in the component. The union is over all components in $\mathcal{C}(A_{j})$. That is,

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Lemma 5

If $3<j\leq n$, then

$P_{2^{n}-1}\setminus A_{j}$* has $2^{n-j}$ components of the form $P_{2^{j-1}-1}$.* 2. 2.

If $\mathcal{C}$ is a component of $P_{2^{n}-1}\setminus A_{j}$ and $f(v)=j-1$ for some $v\in\mathcal{C}$, then $E(v)$ is a set of good edges for $\mathcal{C}$. 3. 3.

$\chi_{r}(P_{2^{n}-1}\cup E_{j})=\chi_{r}(P_{2^{n}-1}).$**

Proof. For this proof we denote by $f$ the $k$-ranking of $P_{2^{n}-1}$. We prove Part 1. Let $u_{1}$ and $u_{2}$ be vertices in $P_{2^{n}-1}$ with $f(u_{1})\geq j$ and $f(u_{2})\geq j$ and if $w$ is a vertex between $u_{1}$ and $u_{2}$ then $f(w)<j$. Since $P_{2^{n}-1}$ has unique optimal ranking, every $2^{j-1}$ vertices there is a vertex with label greater than or equal to $j$ (counting from leftmost vertex). This implies that there is a path, of the form $P_{2^{j-1}-1}$, connecting all vertices between $u_{1}$ and $u_{2}$, not including $u_{1}$ and $u_{2}$. This proves that all components of $P_{2^{n}-1}\setminus A_{j}$ are of the form $P_{2^{j-1}-1}$, and the total number of those components is $\lceil(2^{n}-1)/2^{j-1}\rceil=2^{n-j+1}$.

Proof of Part 2. Let $v$ be a vertex in $\mathcal{C}$ with $f(v)=j-1$. Since $j-1$, is the largest label in $\mathcal{C}$), it easy to see that every path containing edges of $E(v)$ does not contribute to increase the ranking of the $\mathcal{C}$.

Proof of Part 3. Let $G=P_{2^{n}-1}\cup E_{j}$ for some $3<j\leq n$. Let $v_{1}$ and $v_{2}$ be vertices in $G$ with $f(v_{1})=f(v_{2})$, suppose that both vertices are connected by a path $P$. Suppose $v_{1}$ and $v_{2}$ are in the same component $\mathcal{C}\in\mathcal{C}(A_{j})$, then $f(v_{1})=f(v_{2})<j$. If $P$ is a path of $\mathcal{C}$, then by the heredity property from $P_{2^{n}-1}$, there is a vertex in $P$ with a label larger than $f(v_{1})$. We now suppose that $P$ is not a path of $\mathcal{C}$. Let $v$ be the vertex in $\mathcal{C}$ with $f(v)=j-1$. These two last facts and the definition of $E(v)$ imply that $P$ contains an edge in $E(v)$. Thus, there is a vertex in $P$ with a label larger than $f(v_{1})$.

We suppose $v_{1}$ and $v_{2}$ are in different components of $P_{2^{n}-1}\setminus A_{j}$. So, $f(v_{1})=f(v_{2})<j$. By the definition of $G$ and $E_{j}$ we see that any path in $G$ connecting two vertices in different component of $P_{2^{n}-1}\setminus A_{j}$ must have at least one vertex in $A_{j}$. Since vertices in $A_{j}$ have labels larger $j-1$, there is a vertex in $P$ with a label larger than $f(v_{1})$.

We now suppose $v_{1}$ and $v_{2}$ are in $A_{j}$. So, $f(v_{1})=f(v_{2})\geq j$. By definition of $k$-ranking there is vertex $w$ in a subpath of $P_{2^{n}-1}$ that connects those two vertices, with $f(w)>f(v_{1})$. Note that $w\in A_{j}$. Since $w$ does not belong to any of the components in $\mathcal{C}(A_{j})$, any other path connecting those two vertices must contain $w$. Therefore, $w\in P$. This proves Part 3.

Theorem 6

If $n>3$, then

$\chi_{r}(P_{2^{n}-1}\cup\bigcup_{j=4}^{n}E_{j})=\chi_{r}(P_{2^{n}-1})=n$* if and only if $\bigcup_{j=4}^{n}E_{j}$ is the set of good edges for $P_{2^{n}-1}$.* 2. 2.

$\bigcup_{j=4}^{n}E_{j}=E(H_{P}).$** 3. 3.

$\mu(P_{2^{n}-1})=(n-3)2^{n}+4$.

Proof. For this proof we denote by $f$ the standard $k$-ranking of $P_{2^{n}-1}$. We prove of Part 1. The proof that the condition is sufficient is straightforward.

To prove that the condition is necessary we use induction. Let $S(t)$ be the statement

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Lemma 5 Part 3. proves $S(4)$. Suppose the $S(k)$ is true for some $4\leq k<n$. Let

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Let $v_{1}$ and $v_{2}$ be vertices in $G_{1}$ with $f(v_{1})=f(v_{2})$, and suppose that both vertices are connected by a path $P$. Suppose that $v_{1}$ and $v_{2}$ are in the same component $\mathcal{C}\in\mathcal{C}(A_{k+1})$, then $f(v_{1})=f(v_{2})<j$. If $P$ is a path of $\mathcal{C}$, by the heredity property from $G_{0}$, there is a vertex in $P$ with a label larger than $f(v_{1})$.

We suppose that $P$ is not a path of $\mathcal{C}$. Let $v$ the vertex in $\mathcal{C}$ with $f(v)=k$. These two last facts and definition of $E(v)$ imply that $P$ contains an edge in $E(v)$. Therefore, there is a vertex in $P$ with a label larger than $f(v_{1})$.

We suppose $v_{1}$ and $v_{2}$ are in different components of $G_{1}\setminus A_{k+1}$. So, $f(v_{1})=f(v_{2})<k+1$. By definition of $G_{1}$ and $E_{k+1}$ we see that any path in $G_{1}$ connecting two vertices in different component of $G_{1}\setminus A_{k+1}$ has at least one vertex in $A_{k+1}$. Since vertices in $A_{k+1}$ have labels larger than $k$, there is a vertex in $P$ with a label larger than $f(v_{1})$.

We now suppose that $v_{1}$ and $v_{2}$ are in $A_{k+1}$. So, $f(v_{1})=f(v_{2})\geq k+1$. By definition of ranking there is a vertex $w$ in a path of $G_{1}$, that connect those two vertices, with $f(w)>f(v_{1})$. Note that $w\in A_{k+1}$. Since $w$ does not belong to any of the components of $G_{1}\setminus A_{j}$, any other path connecting those two vertices must contain $w$. Therefore, $w\in P$. This proves that $S(k+1)$ is true. Thus, $\bigcup_{j=3}^{n}E_{j}$ is a set of good edges for $P_{2^{n}-1}$.

We now prove that $\bigcup_{i=3}^{n}E_{i}$ is the largest possible set of good edges for $P_{2^{n}-1}$. Suppose that $uv$ is a good edge for $P_{2^{n}-1}$ with $f(v)<f(u)=j$. If the vertices $u$ and $v$ are in the same component of $P_{2^{n}-1}\setminus A_{j+1}$, then is easy to see that $uv\in E_{j+1}$. Note that $j$ is the largest label in each component of $P_{2^{n}-1}\setminus A_{j+1}$. If $u$ and $v$ are in different component of $P_{2^{n}-1}\setminus A_{j+1}$, then $uv$ give rise to a path $P$ connecting $u$ and a vertex $w$ where $w$ and $v$ are in the same component and $f(w)=j$. That is a contradiction, because $f(u)=f(w)=j$ and $P$ does not have label larger than $j$. This proves that $\bigcup_{j=3}^{n}E_{j}$ is the set of good edges of $P_{2^{n}-1}$.

The prove of Part 2. is straightforward from Theorem 4 and Part 1.

Proof of Part 3. It easy to see that the vertex $v_{2^{n-1}}$ of $P_{2^{n}-1}$ has label $n$. That is, $v_{2^{n-1}}$ is the vertex with largest color in $P_{2^{n}-1}$. Therefore, $P_{2^{n}-1}\setminus A_{n}$ has exactly two components. Note that each component is equal to $P_{2^{n-1}-1}$ and that the cardinality of $E(v_{2^{n-1}})$ is $(2^{n}-1)-3$. Since $v_{2^{n-1}}$ is the vertex with the largest color in $P_{2^{n}-1}$, it is easy to see (from proof of Part 1 and the proofs of Lemma 5) that the maximum number of edges that can be added to $P_{2^{n}-1}$ without changing the rank is equal to the maximum number of edges that can be added to each component, $P_{2^{n-1}-1}$, plus all edges in $E(v_{2^{n-1}})$.

Let $a_{n}=\mu(P_{2^{n}-1})$. Then from the previous analysis we have that

[TABLE]

This give rise to the recurrence relation $a_{n}=2a_{n-1}+2^{n}-4$. Therefore, solving the recurrence relation we have that $a_{n}=(n-3)2^{n}+4$. This proves Part 3.

4 Enumeration of the Set of Good Edges for $C_{2^{k}}$

In this section we use the results in the previous section to find the maximum set of edges that may be added to $C_{2^{k}}$ without changing the rank number (good edges).

Suppose that $\Gamma$ represents any of the following graphs; $C_{2^{k}}$, $K_{m_{1},m_{2},\dots,m_{t}}$ or the graph defined by the union of two copies of $K_{n}$ joined by an edge $e$. In this section we give sufficient and necessary conditions to determine whether a set of edges $H$ is a good set for $\Gamma$. For all graphs in this section we count the number of elements in each maximum set of good edges.

We recall that in Figure 2 Part (a) we show the graph $G=C_{2^{4}}\cup H$ where $H$ is the set of all good edges for $G$. So, $\chi_{r}(G)=\chi_{r}(C_{2^{4}})=5$. In Figure 2 Part (b) we show the graph $G^{\prime}=(C_{2^{k}}\setminus\{v_{16}\})\cup H^{\prime}$ where $H^{\prime}$ is the set of all good edges for $G^{\prime}$. Since the graph in Figure 2 Part (b) is equivalent to the graph in Figure 1, Theorem 6 can be applied to this graph. Theorem 8 gives sufficient and necessary conditions to determine whether the graphs $G=C_{2^{k}}\cup H$ and $C_{2^{k}}$ have the same rank number and counts the maximum number of good edges.

Proposition 7

If an edge $e$ is good for $P_{2^{k}-1}$ then $e$ is good for $C_{2^{k}}$.

Proof. Since the standard ranking of $P_{2^{k}-1}$ is contained in the ranking of the cycle $C_{2^{k}}$, and the additional vertex is given the highest label, it follows that if edges are good for the path, they will be good for the cycle.

Let $V:=\{v_{1},v_{2},\dots,v_{2^{k}}\}$ be the set of vertices of $C_{2^{k}}$. Notice that set of edges of $P_{2^{k}-1}$ is $V\setminus\{v_{2^{k}}\}$. We define

[TABLE]

Theorem 8

If $k>3$, then

$\chi_{r}(C_{2^{k}}\cup H_{C})=\chi_{r}(C_{2^{k}})=k+1$* if and only if $H_{C}$ is the set of good edges for $C_{2^{k}}$.* 2. 2.

$\mu(C_{2^{n}})=(n-2)2^{n}+1$.

Proof. To prove Part 1, we first show the condition is sufficient. Suppose $\chi_{r}(C_{2^{k}}\cup H_{C})=\chi_{r}(C_{2^{k}})=k+1$. Suppose $E(H)$ is not a set of good edges. Thus, $E(H)$ contains a forbidden edge, therefore the rank number of $C_{2^{k}}$ is greater than $k+1$. That is a contradiction.

Next we show the condition is necessary. It is known that $\chi_{r}(C_{2^{k}})=k+1$ and that this ranking is unique (up to permutation of the two largest labels) [3]. Let $f$ be a ranking of $C_{2^{k}}$ with $k+1$ labels where $f(v_{2^{k}})=k+1$.

Let $e_{1}=\{v_{2^{k}-1},v_{2^{k}}\}$ and $e_{2}=\{v_{2^{k}},v_{1}\}$ be two edges of $C_{2^{k}}$ and let $H^{\prime}$ be the graph formed by edges of $H$ with vertices in $V^{\prime}=V(H)\setminus\{v_{t}\}=\{v_{1},v_{2},\ldots,v_{2^{k}-1}\}$. Theorem 6 Parts 1 and 2 imply that $E(H^{\prime})$ is a set of good edges for the graph $C_{2^{k}}\setminus\{e_{1},e_{2}\}$ if and only if $\chi_{r}(C_{2^{k}}\setminus\{e_{1},e_{2}\}\cup H^{\prime})=k$ (see Figure 2 Part (b)). Note that $V^{\prime}$ is the set of vertices of $C_{2^{k}}\setminus\{e_{1},e_{2}\}$. The vertices of $C_{2^{k}}\setminus\{e_{1},e_{2}\}\cup H^{\prime}$ have same labels as vertices $V^{\prime}$. Combining this property with $f(v_{2^{k}})=k+1$ gives $\chi_{r}(C_{2^{k}}\cup H^{\prime})=k+1$.

We now prove that $\chi_{r}(C_{2^{k}}\cup H)=k+1$. Let $e$ be an edge in $H\setminus H^{\prime}$. Therefore, the end vertices of $e$ are $v_{2^{k}}$ and $v_{n}$ for some $2\leq n\leq 2^{k}-2$. From the ranking $f$ of a cycle we know that $f(v_{2^{k}})=k+1$ and $f(v_{n})<k+1$. Hence we do not create a new path in $C_{2^{k}}\cup H_{C}$ connecting vertices with labels $k+1$.

Proof of Part 2. Let $W$ be the set of edges of the form $\{v_{2^{n}},v_{i}\}$ for $i=2,3,\ldots,2^{n}-2$. The cardinality of $W$ is $2^{n}-3$. From Proposition 7 we know that all good edges for $P_{2^{n}-1}$ are also good for $C_{2^{n}}$. Therefore, the maximum number of edges that can be added to $C_{2^{n}}$ without changing the rank is equal to maximum number of edges that can be added to $P_{2^{n}-1}$ plus all edges in $W$. Thus, $\mu(C_{2^{n}})=\mu(P_{{2^{n}}-1})+\mid W\mid$. This and Theorem 6 Part 3. imply that $\mu(C_{2^{n}})=(n-3)2^{n}+4+2^{n}-3$. Therefore, $\mu(C_{2^{n}})=(n-2)2^{n}+1$.

Theorem 9

Let $m_{1},m_{2},\ldots,m_{t}$ be positive integers with $m_{1}=\max\{m_{i}\}_{i=1}^{t}$. If $G=K_{m_{1},m_{2},\dots,m_{t}}$ is a multipartite graph, then

any edge connecting two vertices in a part of order $m_{1}$ is forbidden, and 2. 2.

any edge connecting any two vertices in any part of order $m_{i}$ where $i\neq 1$ is good. 3. 3.

$\mu(K_{m_{1},m_{2},\dots,m_{t}})=\sum_{i=2}^{t}\dfrac{(m_{i}-1)m_{i}}{2}.$**

Proof. Let $W=\{w_{1},w_{2},\ldots,w_{m_{1}}\}$ be the set of vertices of the part of $G$ with order $m_{1}$. Let $V=\{v_{2},v_{3},\ldots,v_{r}\}$ be the set of vertices of $G\backslash W$. We consider the function

[TABLE]

To see that $f$ is a minimum ranking of $G$, note that reducing any label violates the ranking property.

Proof of Part 1. Any edge connecting two vertices in $W$ gives rise to a path connecting to vertices with same label.

Proof of Part 2. Any edge connecting two vertices in $V$ does not create any new path with vertices with the same label.

Proof of Part 3. Let $U=\{u_{1},\ldots,u_{m_{s}}\}$ the set of vertices of the part of $K_{m_{1},m_{2},\dots,m_{t}}$ with $m_{s}$ vertices and with $s\not=1$. Let $E_{m_{s}}$ be set of edges of the form $\{v_{i},v_{j}\}$ for $i,j$ in $\{1,2,\ldots m_{s}-2\}$ and $j>i+1$. From the proof of Theorem 9 we know that $E_{m_{s}}$ is a set of good edges of $K_{m_{1},m_{2},\dots,m_{t}}$ for $s=2,3,\ldots,t$ (if $s=1$, then $E_{m_{1}}$ will be a forbidden set). The cardinality of $E_{m_{s}}$ is $(m_{s}-1)m_{s}/2$ for $s=2,3,\ldots,t$. This implies that

[TABLE]

This proves Part 3.

Let $G_{5}$ be the graph defined by the union of two copies of $K_{5}$ joined by an edge $e$. In Figure 3 we show $G_{5}\cup H$ where $H$ is the set of all good edges for $G_{5}$. So, $\chi_{r}(G_{5}\cup H)$ $=$ $\chi_{r}(G_{5})=6$ and $\mu(G_{5})=8$. We generalize this example in Theorem 10.

Theorem 10

Let $G_{n}$ be the union of two copies of $K_{n}$ joined by an edge. Then,

any edge connecting a vertex with highest label in one part with any other vertex in the other part is good. All other edges are forbidden. Moreover, if $H$ is the set of all good edges for $G_{n}$, then $\chi_{r}(G_{n}\cup H)=\chi_{r}(G_{n})=n+1$. 2. 2.

$\mu(G_{n})=2(n-1)$.

Proof. To prove Part 1. we suppose that $G_{n}=K\cup K^{\prime}\cup e$ where $K=K^{\prime}=K_{n}$. Let $W=\{w_{1},w_{2},\ldots,w_{n}\}$ be the set of vertices of $K$ and let $V=\{v_{1},v_{2},\ldots,v_{n}\}$ be the set of the vertices of $K^{\prime}$ and $\{w_{1},v_{n}\}$ the set of vertices of $e$. We consider the function

[TABLE]

It is easy to see that $f$ is a minimum ranking of $G_{n}$ and that $\chi_{r}(G_{n})=n+1$. Let

[TABLE]

and

[TABLE]

We prove that $H=H_{1}\cup H_{2}$ is the set of good edges for $G_{n}$.

From the definition of $f$ we know the labels of the vertices in $K$ are distinct and all of the labels in $K^{\prime}$ are distinct. The combination of these properties and the definition of $f(v_{n})$ implies that if an edge $e$ connects one vertex in $K$ with $v_{n}$ it does not create a new edge connecting two edges with same label. Similarly, if an edge $e$ connects one vertex in $K^{\prime}$ with $w_{n}$ it does not create a path connecting two edges with the same label. This proves that $H$ is the set of good edges for $G_{n}$ and that $\chi_{r}(G_{n}\cup H)=\chi_{r}(G_{n})=n+1$.

Suppose that an edge $e$ connects the vertices $w_{i}\in K$ and $v_{j}\in K^{\prime}$ with $i\leq j\not=n$. The path $w_{j}w_{i}v_{j}$ has two vertices with same label without a larger label in between. The proof of the case $j\leq i\not=n$ is similar. Hence if $e\not\in H$, then $e$ is a forbidden edge.

Proof of Part 2. From proof of Part 1. we can see that $H_{1}$ and $H_{2}$ are the sets of good edges of $G_{n}$ and that the cardinality of each set is $n-1$. So, $\mu(G_{n})=\mid H_{1}\mid+\mid H_{2}\mid=2(n-1)$.

Acknowledgment

The authors are indebted to the referees for their comments and corrections that helped to improve the presentation.