A new graph parameter and a construction of larger graph without increasing radio k-chromatic number

Abstract

For a positive integer $k\ge 2$, the radio k-coloring problem is an assignment L of non-negative integers (colors) to the vertices of a finite simple graph G satisfying the condition $|L(u)-L(v)| \ge k+1-d(u,v)$, for any two distinct vertices u, v of G and d(u, v) being distance between u, v. The span of L is the largest integer assigned by L, while 0 is taken as the smallest color. An $rc_k$-coloring on G is a radio k-coloring on G of minimum span which is referred as the radio k-chromatic number of G and denoted by $rc_k(G)$. An integer h, $0<h<rc_k(G)$, is a hole in a $rc_k$-coloring on G if h is not assigned by it. In this paper, we construct a larger graph from a graph of a certain class by using a combinatorial property associated with $(k-1)$ consecutive holes in any $rc_k$-coloring of a graph. Exploiting the same property, we introduce a new graph parameter, referred as $(k-1)$-hole index of G and denoted by $\rho _k(G)$. We also explore several properties of $\rho _k(G)$ including its upper bound and relation with the path covering number of the complement $G^c$.

1 Introduction

In wireless communication networks, the frequency assignment problem (FAP) is a well-known problem which deals with the assignment of frequencies (non-negative integers) to transmitters in an optimal manner such that the potential risk of interference, due to proximity of two transmitters having same or nearly same frequencies, is averted. The radio k-coloring problem, introduced by Chartrand et al. (2005) is a graph-theoretic variant of FAP where the vertices represent the transmitters and the edges are assigned according to the proximity of the transmitters. For any positive integer $k\ge 2$, a radio k-coloring L of a finite simple graph G is a mapping $L : V \rightarrow {\mathbb {N}}\cup \{0\}$ such that for any two vertices u, v in G,

$$\begin{aligned} |L(u)-L(v)| \ge k+1-d(u,v) \end{aligned}$$

(1)

The span of a radio k-coloring L is defined to be $( max_{v\in V}L(v)-min_{v\in V}L(v) )$ and is denoted by span(L). The radio k-chromatic number of G, denoted by $rc_k(G)$, is defined to be $min_L\{span(L)$ : L is a radio k-coloring of G}. Without loss of generality, we shall assume $min_{v\in V}L(v)=0$ for any radio k-coloring L on G. Any radio k-coloring L on G with span $rc_k(G)$ is referred as $rc_k(G)$-coloring or simply $rc_k$-coloring (when the underlying graph is fixed). An integer $h,~0< h < rc_k(G)$, is said to be a hole in a $rc_k$-coloring L on G if it is not assigned as a color to any vertex of G by L.

So far the radio k-coloring problem has been mostly studied for $k =diam(G)$, $k =diam(G)-1$, $k =diam(G)-2$, $k=3$, $k=2$. For $k=2$, the problem becomes the L(2, 1)-coloring problem introduced by Griggs and Yeh (1992). Also note that $rc_2(G)$ is denoted as $\lambda _{2,1}(G)$ and $rc_2$-coloring of G is referred as $\lambda _{2,1}$-coloring of G. Any graph G which admits a $\lambda _{2,1}$-coloring without any hole is said to be full colorable.

The L(2, 1)-coloring problem has received extensive attention. Readers may find an extensive survey in Calamoneri (2011). Several interesting combinatorial properties of holes of a $\lambda _{2,1}$-coloring as well as existence of full colorable graphs have been studied in Adams et al. (2007), Fishburn and Roberts (2006) and Georges and Mauro (2005). On the other hand, for $k\ne 2$, the radio k-coloring problem has been studied for very few limited families of graphs including paths, trees and cycles (Chartrand et al. 2000; Khennoufa and Togni 2005; Li et al. 2010; Liu 2008), powers of paths and cycles (Liu and Xie 2004, 2009; Saha and Panigrahi 2012), toroidal grids (Saha and Panigrahi 2013) etc. Also lower bound for radio k-chromatic number has been studied in Saha and Panigrahi (2015). But no significant attempt has been made so far to understand the behaviour of holes in any $rc_k$-coloring, up to the best of our knowledge. Besides, constructing a larger graph $G^\prime$ containing a given graph G with $rc_k(G^\prime )=rc_k(G)$ is also an interesting combinatorial problem. Given a $rc_k$-coloring of G, this problem becomes even more intriguing if a $rc_k$-coloring of $G^\prime$ is required to maintain the previous assignment of colors to the vertices of G.

2 Perspective and our contribution

Due to rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, an important task is to construct an extended network (connected graph) from a given network (graph) such that (i) the frequency (colors) allocation scheme to the old stations (vertices) remains same and (ii) the unused frequencies for the old system can be assigned to the new stations. To address this issue, we study an interesting combinatorial property of consecutive holes of an $rc_k$-coloring of a graph. Based on the fact that no $rc_k$-coloring on G contains k consecutive holes (Sarkar and Adhikari 2015), we explore the structural properties of graphs whose every $rc_k$-coloring has $(k-1)$ consecutive holes and show that if such a graph G belongs to a certain class then G is an induced subgraph of a graph $G^*$ in the same class such that $rc_k(G^*)=rc_k(G)$ and at least one $rc_k$-coloring on $G^*$ contains at most $(k-2)$ consecutive holes. While describing an $rc_k$-coloring of $G^*$ with such desired feature, we use the colors of an $rc_k$-coloring of G as well as some of its holes. As the holes in an $rc_k$-coloring are unused frequencies in between the allotted spectrum, this enables us to think of an expanded network whose frequency assignment is done by already used as well as some unused frequencies of the smaller network.

We refer the minimum number of occurrence of $(k-1)$ consecutive holes in any $rc_k$-coloring on G as the $(k-1)$-hole index of G and denote by $\rho _{k}(G)$ or simply $\rho _{k}$ if there is no confusion regarding the graph. We show $\varDelta (G)$, the maximum degree of a vertex in G, is a general upper bound of $\rho _{k}(G)$ and study the structure of G when $\rho _k(G)=\varDelta (G)$ and $\rho _k(G)=\varDelta (G)-1$ for any $k\ge 2$. We also give another class such that for any graph G with n vertices in the class, if $rc_k(G)\ge (n-1)(k-1)$ then $\rho _k(G)=c(G^c)-1$ and if $c(G^c)\ge 2$ then $\rho _{k_1}(G)=\rho _{k_2}(G)$, for $k_1,~k_2\ge 2$, where $c(G^c)$ is the path covering number of the complement $G^c$ i.e., minimum number of vertex disjoint paths required to exhaust all vertices of $G^c$. These findings eventually extend the works of Georges and Mauro (2005) on hole index, i.e., the minimum number of holes in any $rc_2$-coloring, of any graph G and its similar relations with $\varDelta (G)$ and $c(G^c)$. We further extend the notion of the island sequence (Georges and Mauro 2005), which is a non-decreasing sequence of cardinalities of maximal sets of consecutive integers used by an $rc_2$-coloring on G with minimum holes, in a general perspective of radio k-coloring for any $k\ge 2$ and prove some interesting results.

3 Preliminaries

Throughout this paper, unless otherwise stated, graphs are taken as finite and simple with at least two vertices and we assume $k\ge 2$. The vertex set of a graph G is denoted by V(G). For any $rc_k$-coloring L on G, let $L_i= \{v\in V(G) |~L(v)=i\}$ and $l_i=|L_i|$.

For definitions of disjoint union and Cartesian product of graphs (denoted by $G+H$ and $G\square H$ respectively, for two graphs G and H), Hamiltonian path, domination number (denoted by $\gamma (G)$), radius and centre of G (denoted by rad G and C(G) respectively) the reader is referred to West (2001). The disjoint union of l copies of G is denoted by lG. A set $S\subset V(G)$ is said to be a perfect dominating set if every vertex in $V(G)\setminus S$ is adjacent to exactly one vertex in S (see Haynes et al. 1998). The minimum cardinality of a perfect dominating set of G is the perfect domination number of G, denoted by $\gamma _p(G)$. Clearly $\gamma (G)\le \gamma _p(G)$. The complete graph, cycle and path with n vertices are denoted as $K_n$, $C_n$ and $P_n$ respectively.

We now define a new class of graphs for which we will show later that given any graph G in that class, either $\rho _k(G)=0$ or G is an induced subgraph of $G^{\prime }$ in the same class such that $rc_k(G^{\prime })=rc_k(G)$ and $\rho _k(G^{\prime })=0$ (see Theorem 4). Let ${\mathcal {H}}_k$ be the class of simple finite graphs such that $G\in {\mathcal {H}}_k$ if and only if for every edge e in G, there exists a vertex $\omega _e \in V(G)$ which is at a distance $\lfloor \frac{k}{2}\rfloor$ from one vertex incident to e and at a distance $\lceil \frac{k}{2}\rceil +1-k(mod~2)$ from the other vertex incident to e.

Remark 1

Note that ${\mathcal {H}}_k\subseteq {\mathcal {H}}_{k-1}$ for $k\ge 3$.

Explanation: Let $G\in {\mathcal {H}}_k$. Also let $e=(x,~y)$ be any edge in G.

Case I   Let $k=2r$ and $\omega _e\in V(G)$ be such that $d(x,~\omega _e)=r$ and $d(y,~\omega _e)=r+1$. Let $\omega$ be the vertex immediately before $w_e$ on a shortest path from x to $\omega _e$. Then $d(x,~\omega )=r-1$ and $d(y,~\omega )=r$. Since e is any edge in G, $G\in {\mathcal {H}}_{2r-1}={\mathcal {H}}_{k-1}$.

Case II   Let $k=2r+1$ and $\omega _e\in V(G)$ be such that $d(x,~\omega _e)=r$ and $d(y,~\omega _e)=r+1$. Since e is any edge in G, $G\in {\mathcal {H}}_{2r}={\mathcal {H}}_{k-1}$.

Examples of ${\mathcal {H}}_k$:

• (i) $C_n\in {\mathcal {H}}_k$, where $n\ge 4$ and $2\le k\le n-1-n~(mod~2)$.

• (ii) $C_n\square P_m\in {\mathcal {H}}_k$, where $n\ge 4$, $m\ge 2$ and $2\le k\le n-1-n~(mod~2)+m-m~(mod~2)$.

• (iii) Let T be a tree. Then $T\in {\mathcal {H}}_k$, where $2\le k\le 2_\cdot rad~T+(-1)^{|C(T)|-1}$ and $rad~T=min_{u\in V(T)}\{max_{v \in V(T)}d(u,~v)\}$. Also |C(T)| is the number of vertices u of T such that $max_{v \in V(T)}d(u,~v)$ is minimum.

• (iv) The graph in Figure 1 is an example of ${\mathcal {H}}_k$, for $2\le k\le 7$.

Fig. 1

Illustrating an example of ${\mathcal {H}}_k$, for $2\le k\le 7$

Also let ${\mathcal {G}}_1$ be the class of triangle-free graphs and ${\mathcal {G}}_2$ be the family of graphs with a Hamiltonian path in every component of their complements. A complete bipartite graph, the Cayley graph $G=Cay({\mathbb {Z}}_n, S)$, where $S={\mathbb {Z}}_n\setminus \{\overline{0}, \overline{m}, -\overline{m}\}$ such that $gcd(m,n)>1$, are some examples of ${\mathcal {G}}_2$. For these two classes, the following results will help us later to get some interesting properties of $\rho _k(G)$ (see Theorem 9 and Corollary 3).

Theorem 1

(Sarkar and Adhikari 2015) Let $G\in {\mathcal {G}}_1\cup {\mathcal {G}}_2$ be a graph with n vertices. Then

• (i) $rc_k(G)\le (n-1)(k-1)$ if and only if $G^c$ has a Hamiltonian path.

• (ii) $rc_k(G) = n(k-1)+r-k$ if and only if $c(G^c) = r$, when $r\ge 2$.

Beside this, the following lower bound for domination number of graph will be used in the proof of Theorem 7.

Theorem 2

(Haynes et al. 1998; Walikar et al. 1979) For any graph G of order n, $\gamma (G)\ge \lceil \frac{n}{\varDelta (G)+1}\rceil$.

4 Preparatory definitions and results

We start with the following definitions.

Definition 1

A $(k-1)$-hole in a $rc_k$-coloring L on a graph G is a sequence of $(k-1)$ consecutive holes in L. If L has $(k-1)$ consecutive holes $i+1$, $i+2,\ldots$, $i+k-1$, then the corresponding $(k-1)$-hole is denoted by $\{i+1,i+2,\ldots ,i+k-1\}$.

Remark 2

The minimum number of $(k-1)$-holes in any $rc_k$-coloring on G is the $(k-1)$-hole index of G. We refer the collection of all $rc_k$-colorings on G with $\rho _k(G)$ number of $(k-1)$-holes as $\varLambda _{\rho _k}(G)$.

Definition 2

The minimum span of a radio k-coloring on G with at most $(k-2)$ consecutive holes is defined as max-$(k-2)$-hole span of G and denoted by $\mu _k(G)$.

Remark 3

It is easy to observe that $rc_k(G)\le \mu _k(G)$, for any graph G. Clearly if $rc_k(G)=\mu _k(G)$, then $\rho _k(G)=0$.

Definition 3

Let $L\in \varLambda _{\rho _k}(G)$. An $\varOmega$-set of L is a non-empty set A of non-negative integers, assigned by L, such that $s\in A$ only if $0\le |s-s^\prime |<k$, for some $s^\prime \in A$. A maximal $\varOmega$-set of $L\in \varLambda _{\rho _k}(G)$ is a k-island or simply an island of L. The minimum and maximum element of an island I are said to be the left and right coast of I respectively, denoted by lc(I) and rc(I) accordingly. The left and right coasts of I are together called the coastal elements of I.

We now prove some results which will be useful in proving our main results.

Lemma 1

For $k\ge 2$, let L be an $rc_k$-coloring of a graph G with a $(k-1)$-hole $\{i+1,~i+2,\ldots ,~i+k-1\}$, where $i\ge 0$. Then there are two vertices $u\in L_i$ and $v\in L_{i+k}$ such that u and v are adjacent in G.

Proof

If possible, let no vertex of $L_i$ be adjacent to any vertex in $L_{i+k}$ in G. Define a new radio k-coloring $\hat{L}$ given by

$$\begin{aligned} \hat{L}(u)=\left\{ \begin{array} {ll} L(u), &{}\quad \text{ if }\, L(u)\le i, \\ L(u)-1, &{}\quad \text{ if } \, L(u)\ge i+k. \end{array}\right. \end{aligned}$$

Since no vertex of $\hat{L}_i$ is adjacent to a vertex of $\hat{L}_{i+k-1}$ in G and L is a proper radio k-coloring of G, $\hat{L}$ is a proper radio k-coloring of G with span $rc_k(G)-1$, a contradiction. This completes the proof. $\square$

In the next result, we consider G satisfies $\rho _k(G)\ge 1$.

Lemma 2

Let G be a graph with $\rho _k(G)\ge 1$ and $L\in \varLambda _{\rho _k}(G)$. If $\{i+1,~i+2,\ldots ,~i+k-1\}$ is a $(k-1)$-hole in L, then $l_{i}=l_{i+k}$ and the subgraph of G induced by $L_{i}\cup L_{i+k}$ is $l_{i}K_2$.

Proof

$\mathbf{Case I }$ Let $l_{i+k}\ge 2$. If possible, let $x\in L_{i+k}$ be such that x is not adjacent to any vertex in $L_i$. Define $\hat{L}$ by

$$\begin{aligned} \hat{L}(u)=\left\{ \begin{array}{ll} L(u), &{}\quad \text{ if } \, u\ne x,\\ i+k-1, &{}\quad \text{ if } \, u=x. \end{array}\right. \end{aligned}$$

Hence $\hat{L}$ is an $rc_k$-coloring with fewer $(k-1)$-holes, leading to a contradiction.

$\mathbf Case II$ Let $l_{i+k}=1$ and $L_{i+k}=\{x\}$. If possible, let x be not adjacent to any vertex in $L_i$. Define $\hat{L}$ by

$$\begin{aligned} \hat{L}(u)=\left\{ \begin{array}{ll} L(u), &{}\quad \text{ if } \, L(u)\le i,\\ L(u)-1, &{}\quad \text{ if } \, L(u)\ge i+k. \end{array}\right. \end{aligned}$$

Clearly, $\hat{L}$ is a radio k-coloring with span $rc_k(G)-1$, a contradiction.

The above two cases suggest that each vertex of $L_{i+k}$ is adjacent to some vertex in $L_i$. Also no two vertices in $L_{i+k}$ can be adjacent to the same vertex in $L_i$. Hence $l_{i}\ge l_{i+k}$.

Now $l_{i+k}\ge 1$. If $l_{i}=1$, we are done. Let $l_{i}\ge 2$. Then by a similar argument as before we can show each vertex $u\in L_i$ is adjacent to a unique vertex in $L_{i+k}$. Therefore $l_{i+k}\ge l_i$. Hence we have $l_{i}=l_{i+k}$.

Since no two vertices in $L_i$ can be adjacent to the same vertex in $L_{i+k}$ in G and vice versa, the subgraph of G induced by $L_{i}\cup L_{i+k}$ is $l_{i}K_2$.

This completes the proof. $\square$

4.1 A recoloring from a coloring in $\varLambda _{\rho _k}(G)$ and an equivalence relation

Let $L\in \varLambda _{\rho _k}(G)$ and $I_0,~I_1,\ldots ,~I_{\rho _k}$ be the islands of L such that $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$. For any i, $0\le i\le \rho _k$, and m, $0\le m\le (\rho _k-i)$, we define a new coloring $\hat{L}$ on G given by

Clearly $\hat{L}\in \varLambda _{\rho _k}(G)$ and the islands of $\hat{L}$ are $I_0^{\prime },~I_1^{\prime },\ldots ,~I_{\rho _k}^{\prime }$ such that $I_j^{\prime }=I_j$, for $j\notin \{i,~i+1,\ldots ,~i+m\}$ and $rc(I_{i+m-t}^{\prime })=rc(I_{i+m})-(lc(I_{i+t})-lc(I_i))$, $lc(I_{i+m-t}^{\prime })=rc(I_{i+m})-(rc(I_{i+t})-lc(I_i))$, for $0\le t\le m$. We refer this recoloring $\hat{L}$ as an $\alpha$-recoloring of L.

Define a relation $\eta$ on $\varLambda _{\rho _k}(G)$ given by $L_1\eta L_2$ if and only if $L_2$ is obtained from $L_1$ by applying a finite number of suitable $\alpha$-recolorings described above. Then $\eta$ is an equivalence relation. Hence every radio k-coloring in $\varLambda _{\rho _k}(G)$ is $\eta$-related to some $L\in \varLambda _{\rho _k}(G)$ which has islands $I_0,~I_1,\ldots ,~I_{\rho _k}$ with $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$ such that $|I_0|\le |I_1|\le \cdots \le |I_{\rho _k}|$. Thus we assume, without loss of generality, that the islands $I_0,~I_1,\ldots ,~I_{\rho _k}$ with $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$, of any $L\in \varLambda _{\rho _k}(G)$ satisfy the inequalities $|I_0|\le |I_1|\le \cdots \le |I_{\rho _k}|$ and we refer the finite sequence $(|I_0|,~|I_1|,\ldots ,~|I_{\rho _k}|)$ as island sequence of L.

Corollary 1

Let G be a graph with $\rho _k(G)\ge 1$ and I,  J be two distinct islands of $L\in \varLambda _{\rho _k}(G)$ with x and y as two coastal elements of I and J respectively. Then $l_{x}=l_{y}$ and the subgraph of G induced by $L_{x}\cup L_{y}$ is $l_{x}K_2$.

Proof

Considering some suitable $\alpha$-recolorings, we can assume, without loss of generality, that $x=rc(I)$, $y=lc(J)$ and $y=x+k$. The proof follows from Lemma 2. $\square$

Lemma 3

Let G be a graph with $\rho _k(G)\ge 1$ and $\rho _k(G)=\varDelta (G)-r$. Also let $L\in \varLambda _{\rho _k}(G)$ which has islands $I_0,~I_1,\ldots ,~I_{\rho _k}$ with $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$ such that $|I_0|\le |I_1|\le \cdots \le |I_{\rho _k}|$. Then $|I_j|=1$, where $0\le j\le \varDelta (G)-2r$.

Proof

If possible, let $|I_{\varDelta (G)-2r}|\ge 2$. Then for every j, $\varDelta (G)-2r\le j\le \varDelta (G)-r$, $lc(I_j)\ne rc(I_j)$ and these islands have total $2(r+1)$ number of coastal elements. Let u be a vertex in G such that $L(u)=lc(I_0)=0$. Then by Corollary 1, u is adjacent to at least one vertex from each of $I_j$, $1\le j\le \varDelta (G)-2r-1$ and at least two vertices from each of $I_j$, $\varDelta (G)-2r\le j\le \varDelta (G)-r$ in G. Hence $d(u)\ge (\varDelta (G)-2r-1)+2(r+1)=\varDelta (G)+1$, a contradiction.$\square$

5 Main results

5.1 Larger graph with the same radio k-chromatic number

In this subsection, we obtain larger graph with reduced $(k-1)$-hole index while keeping the radio k-chromatic number unchanged.

Theorem 3

Let $G\in {\mathcal {H}}_k$ with $\rho _k(G)\ge 1$. Then there is a graph $G^*\in {\mathcal {H}}_k$ such that

• (i) G is an induced subgraph of $G^*$;

• (ii) $rc_k(G^*)=rc_k(G)$;

• (iii) $\rho _k(G^*)=\rho _k(G)-1$.

Proof

Let L be any $rc_k$-coloring on G. Since $\rho _k(G)\ge 1$, L has at least one $(k-1)$-hole. Let $\{i+1,~i+2,\ldots ,~i+k-1\}$ be a $(k-1)$-hole in L where i is an non-negative integer. By Lemma 1, there are two vertices $x\in L_i$ and $y\in L_{i+k}$ so that $e=(x,~y)$ is an edge in G.

$\mathbf Case I ~~$ Let $\omega _e$ be the vertex in G such that $d(x,~\omega _e)=\lfloor \frac{k}{2}\rfloor$ and $d(y,~\omega _e)=\lceil \frac{k}{2}\rceil +1-k(mod~2)$. We construct a new graph $G^*$ by adding a new vertex $\omega ^*$ and a new edge $(\omega ^*,~\omega _e)$ to G. Note that $G^*\in {\mathcal {H}}_k$, since $G\in {\mathcal {H}}_k$ and $x=\omega _{e^\prime }$, where $e^\prime =(\omega _e,~\omega ^*)$. Also G is an induced subgraph of $G^*$. Define a new coloring $L^*$ on $G^*$ given by

$$\begin{aligned} L^*(u)=\left\{ \begin{array} {ll} L(u), &{}\quad \text{ if } \, u\in V(G) \\ i+\lceil \frac{k}{2}\rceil , &{}\quad \text{ if } \, u=\omega ^*\end{array}\right. \end{aligned}$$

We shall show that $L^*$ is a proper radio k-coloring on $G^*$.

Now since $i+1,~i+2,\ldots ,~i+k-1$ are consecutive holes in L, either

$$\begin{aligned} L^*(\omega _e)\ge L^*(y)+k+1-d(\omega _e,~y)=i+k+\lfloor \frac{k}{2}\rfloor +k(mod~2) \end{aligned}$$

(2)

or

$$\begin{aligned} L^*(\omega _e)\le L^*(x)-(k+1-d(\omega _e,~x))=i-\lceil \frac{k}{2}\rceil -1 \end{aligned}$$

(3)

From Eq. (2), $L^*(\omega _e)-L^*(\omega ^*)\ge k$ and from Eq. (3), $L^*(\omega ^*)-L^*(\omega _e)> k$. Thus

$$\begin{aligned} |L^*(\omega ^*)-L^*(\omega _e)|\ge k \end{aligned}$$

(4)

Again

$$\begin{aligned} |L^*(\omega ^*)-L^*(x)|=k+1-d(\omega ^*,~x) \end{aligned}$$

(5)

and

$$\begin{aligned} |L^*(\omega ^*)-L^*(y)|\ge k+1-d(\omega ^*,~y) \end{aligned}$$

(6)

Let $u\in V(G^*)\setminus \{x,~y,~\omega _e,~\omega ^*\}$ be in the same connected component of $G^*$ containing $\omega ^*$.

$\mathbf Case I \mathrm{(a)}~~$ Let $d(x,~u)=k-r_1$ and $d(y,~u)=k-r_2$, where $0\le r_1,~r_2\le k-1$.

Now, $d(x,~u)=k-r_1\Rightarrow |L^*(u)-L^*(x)|\ge r_1+1$. This implies either $L^*(u)\le i-r_1-1$ or $L^*(u)\ge i+r_1+1$, i.e., either $L^*(u)\le i-r_1-1$ or $L^*(u)\ge i+k$, since $0\le r_1\le k-1$ and $i+1,~i+2,\ldots ,~i+k-1$ are holes in L.

Again, $d(y,~u)=k-r_2\Rightarrow |L^*(u)-L^*(y)|\ge r_2+1$. This implies either $L^*(u)\ge i+k+r_2+1$ or $L^*(u)\le i+k-r_2-1$, i.e., either $L^*(u)\ge i+k+r_2+1$ or $L^*(u)\le i$ since $0\le r_2\le k-1$ and $i+1,~i+2,\ldots ,~i+k-1$ are holes in L.

As $d(x,~u)=k-r_1$ and $d(y,~u)=k-r_2$, $0\le r_1,~r_2\le k-1$, so considering the above two situations, we get either $L^*(u)\le i-r_1-1$ or $L^*(u)\ge i+k+r_2+1$.

Now let $L^*(u)\le i-r_1-1$. Then $L^*(\omega ^*)-L^*(u)\ge \lceil \frac{k}{2}\rceil +r_1+1$.

Also, $d(\omega _e,~u)\ge d(x,~u)-d(\omega _e,~x)=\lceil \frac{k}{2}\rceil -r_1$. So, $d(\omega ^*,~u)\ge \lceil \frac{k}{2}\rceil +1-r_1$ implying $k+1-d(\omega ^*,~u)\le \lfloor \frac{k}{2}\rfloor +r_1$. Hence we have $L^*(\omega ^*)-L^*(u)>k+1-d(\omega ^*,~u)$. Therefore $|L^*(\omega ^*)-L^*(u)|>k+1-d(\omega ^*,~u)$, if $L^*(u)\le i-r_1-1$.

Consider the other way, i.e., let $L^*(u)\ge i+k+r_2+1$. Then $L^*(u)-L^*(\omega ^*)\ge \lfloor \frac{k}{2}\rfloor +r_2+1$.

But $d(\omega _e,~u)\ge d(y,~u)-d(\omega _e,~y)\Rightarrow d(\omega ^*,~u)\ge \lceil \frac{k}{2}\rceil -r_2\Rightarrow k+1-d(\omega ^*,~u)\le \lfloor \frac{k}{2}\rfloor +r_2+1$. Hence we have $L^*(u)-L^*(\omega ^*)\ge k+1-d(\omega ^*,~u)$, i.e., $|L^*(u)-L^*(\omega ^*)|\ge k+1-d(\omega ^*,~u)$, if $L^*(u)\ge i+k+r_2+1$.

Therefore

$$\begin{aligned} |L^*(\omega ^*)-L^*(u)|\ge k+1-d(\omega ^*,~u), \end{aligned}$$

(7)

whenever $u\in V(G^*)\setminus \{x,~y,~\omega _e,~\omega ^*\}$ with $d(x,~u)=k-r_1$ and $d(y,~u)=k-r_2$, $0\le r_1,~r_2\le k-1$.

$\mathbf Case I \mathrm{(b)}~~$ Let $d(x,~u)=k$ and $d(y,~u)=k+1$. Then $L^*(u)\le i-1$ or $L^*(u)\ge i+k$. Also, $d(\omega _e,~u)\ge d(x,~u)-d(\omega _e,~x)=\lceil \frac{k}{2}\rceil$. So, $k+1-d(\omega ^*,~u)\le \lfloor \frac{k}{2}\rfloor \le |L^*(\omega ^*)-L^*(u)|$.

Since x and y are adjacent in G, $|d(x,~u)-d(y,~u)|\le 1$. Hence $\textit{Case I.(b)}$, together with equation (7) of $\textit{Case I.(a)}$, implies

$$\begin{aligned} |L^*(\omega ^*)-L^*(u)|\ge k+1-d(\omega ^*,~u), \end{aligned}$$

(8)

whenever $u\in V(G^*)\setminus \{x,~y,~\omega _e,~\omega ^*\}$ with $d(x,~u)=k-r$, $0\le r\le k-1$.

Now the only remaining possibility of $\textit{Case I}$ is the following.

$\mathbf Case I$(c) Let $d(x,~u)=k+r$, $r\ge 1$. Then $d(y,~u)\ge k+r-1\ge k$. Therefore $|L^*(\omega ^*)-L^*(u)|\ge \lfloor \frac{k}{2}\rfloor$.

Now, $d(\omega _e,~u)\ge d(x,~u)-d(\omega _e,~x)=\lceil \frac{k}{2}\rceil +r$. So, $k+1-d(\omega ^*,~u)\le \lfloor \frac{k}{2}\rfloor -r< |L^*(\omega ^*)-L^*(u)|$, as $r\ge 1$. Thus

$$\begin{aligned} |L^*(\omega ^*)-L^*(u)|> k+1-d(\omega ^*,~u), \end{aligned}$$

(9)

whenever $u\in V(G^*)\setminus \{x,~y,~\omega _e,~\omega ^*\}$ with $d(x,~u)=k+r$, $r\ge 1$.

The equations (4), (5), (6), (8) and (9) together suggest that $L^*$ is a proper radio k-coloring of $G^*$.

$\mathbf Case II ~~$ Let $\omega _e$ be a vertex in G such that $d(y,~\omega _e)=\lfloor \frac{k}{2}\rfloor$ and $d(x,~\omega _e)=\lceil \frac{k}{2}\rceil +1-k(mod~2)$. Similar to Case I, we construct a new graph $G^*$ by adding a new vertex $\omega ^*$ and a new edge $(\omega ^*,~\omega _e)$ to G. Note that $G^*\in {\mathcal {H}}_k$, since $G\in {\mathcal {H}}_k$ and $y=\omega _{e^\prime }$, where $e^\prime =(\omega _e,~\omega ^*)$. Also G is an induced subgraph of $G^*$. Define a new coloring $\hat{L}$ on $G^*$ by

$$\begin{aligned} \hat{L}(u)=\left\{ \begin{array} {ll} L(u), &{}\quad \text{ if } \, u\in V(G) \\ i+\lfloor \frac{k}{2}\rfloor , &{}\quad \text{ if } \, u=\omega ^*\end{array}\right. \end{aligned}$$

Similar argument, as in the former case, shows $\hat{L}$ is a proper radio k-coloring of $G^*$.

As in both cases, G is an induced subgraph of $G^*$ and $G^*$ admits a radio k-coloring of span $rc_k(G)$, so $rc_k(G^*)=rc_k(G)$.

Since L is an arbitrary $rc_k$-coloring on G, applying the above argument for any $L\in \varLambda _{\rho _k}(G)$, we get $\rho _k(G^*)\le \rho _k(G)-1$. Again since G is an induced subgraph of $G^*$ which has exactly one vertex more than G, $\rho _k(G^*)\ge \rho _k(G)-1$. Hence $\rho _k(G^*)=\rho _k(G)-1$. This completes the proof.$\square$

Theorem 4

Let $G\in {\mathcal {H}}_k$ be a graph with $\rho _k(G)\ge 1$. Then there exists a graph $G^*\in {\mathcal {H}}_k$, such that

• (i) G is an induced subgraph of $G^*$;

• (ii) $rc_k(G^*)=rc_k(G)$;

• (iii) $\rho _k(G^*)=0$.

Proof

By repeated use of Theorem 3, the proof follows immediately.$\square$

Recall that $\mu _k(G)$ is the minimum span of any radio k-coloring on G with at most $(k-2)$ consecutive holes. We now prove the following result.

Theorem 5

Let n be a positive integer such that $n=rc_k(G)$, for some $G\in {\mathcal {H}}_k$. Then there exists a graph $G^*\in {\mathcal {H}}_k$, containing G as its induced subgraph, such that $rc_k(G^*)=n$ and $\mu _k(G^*)=rc_k(G^*)$.

Proof

If $\rho _k(G)=0$ then $G^*=G$, otherwise the proof follows from Theorem 4.$\square$

We now prove an interesting result for L(2,  1)-coloring. Recall that for any graph G, $rc_2(G)=\lambda _{2,1}(G)$ and G is full colorable if $\rho _2(G)=0$.

Corollary 2

Let n be a positive integer such that $n=\lambda _{2,1}(G)$, for some $G\in {\mathcal {H}}_2$. Then there exists a graph $G^*\in {\mathcal {H}}_2$, such that

• (i) G is an induced subgraph of $G^*$;

• (ii) $\lambda _{2,1}(G^*)=n$;

• (iii) $G^*$ is full colorable.

Proof

If $\rho _2(G)=0$ then $G^*=G$, otherwise the proof follows from Theorem 4.$\square$

5.2 A general upper bound of $\rho _k(G)$ and related results

We begin with an upper bound of $\rho _k(G)$.

Theorem 6

Let G be a graph with $\rho _k(G)\ge 1$. Then $\rho _k(G)\le \varDelta (G)$.

Proof

Let $L\in \varLambda _{\rho _k}(G)$ which has islands $I_0,~I_1,\ldots ,~I_{\rho _k}$. Let x be a coastal element of $I_0$ and $u\in L_x$. Then u is adjacent to some $v_y\in L_y$ in G, for each coastal element y of $I_j$, $1\le j\le \rho _k(G)$ by Corollary 1. Hence $d(u)\ge \rho _k(G)$. Therefore $\rho _k(G)\le \varDelta (G)$. $\square$

We now explore the structure of G when $\rho _k(G)$ attains its upper bound.

Theorem 7

Let G be a graph of order n with $\rho _k(G)\ge 1$ and $\rho _k(G)=\varDelta (G)=\varDelta$. Then

• (i) G is a $\varDelta$-regular graph;

• (ii) $rc_k(G)=k\varDelta$;

• (iii) $n\equiv 0$ $(\text{ mod }~\varDelta +1)$;

• (iv) $\gamma (G)=\gamma _p(G)=\frac{n}{\varDelta +1}$;

• (v) If $n\ne \varDelta +1$, then $\mu _k(G)=rc_k(G)+1$.

Proof

Let $L\in \varLambda _{\rho _k}(G)$ which has islands $I_0,~I_1,\ldots ,~I_{\rho _k}$ with $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$ such that $|I_0|\le |I_1|\le \ldots \le |I_{\rho _k}|$. Then L has $\rho _k+1$ number of islands.

If possible, let I be an island of L such that $lc(I)\ne rc(I)$. Then by Corollary 1, for any vertex v with L(v) as a coastal element of any other island J of L, $d(v)\ge \rho _k+1=\varDelta +1$, a contradiction. Hence for every island I of L, $lc(I)=rc(I)$ i.e., the islands are singleton sets. Thus, for every vertex v of G, L(v) is a coastal element of some island of L and so degree of each vertex is same, by Corollary 1. Hence G is $\varDelta$-regular.

As the islands are singleton sets, so $I_j=\{kj\}$, $0\le j\le \varDelta$. Hence $rc_k(G)=k\varDelta$.

Again, by Corollary 1, $l_{ki}=l_{kj}=l$ (say), $0\le i<j\le \varDelta$. Hence $n=l(\varDelta +1)$, i.e., $n\equiv 0$ $(\text{ mod }~\varDelta +1)$.

Since the subgraph induced by $L_{ki}\cup L_{kj}$ is $lK_2$, for any i,  j with $0\le i<j\le \varDelta$ by Corollary 1 and $I_i=\{ki\}$, $0\le i\le \varDelta$, $L_{ki}$ is a dominating set as well as a perfect dominating set of G, for every i, $0\le i\le \varDelta$. Hence $\gamma (G)\le \frac{n}{\varDelta +1}$. Using Theorem 2, we get $\gamma (G)=\frac{n}{\varDelta +1}$. Hence $\gamma _p(G)=\frac{n}{\varDelta +1}$.

Let $n\ne \varDelta +1$. Then $l\ge 2$. Since $lc(I_j)=rc(I_j)$, for every $0\le j\le \varDelta$, using Corollary 1, we get a path $P:~v_0,~v_1,\ldots ,~v_\varDelta$ in G such that $L(v_j)\in I_j=\{kj\}$, $0\le j\le \varDelta$. Define a new coloring $\hat{L}$ on G by

$$\begin{aligned} \hat{L}(u)=\left\{ \begin{array} {ll} L(u), &{}\quad \text{ if } \, u\ne v_j,~0\le j\le \varDelta , \\ L(u)+1, &{}\quad \text{ if } \, u=v_j,~0\le j\le \varDelta .\end{array}\right. \end{aligned}$$

Note that $\hat{L}$ is a radio k-coloring of G without $(k-1)$ consecutive holes. Therefore $\mu _k(G)\le span(\hat{L})=rc_k(G)+1$. But since $\rho _k(G)>0$, $\mu _k(G)>rc_k(G)$. Hence $\mu _k(G)=rc_k(G)+1$.$\square$

There may be graph G for which $\rho _k(G)<\varDelta (G)$ with $rc_k(G)=k\varDelta (G)$ for some $k\ge 2$. For example, consider $G=K_n+K_1$, $n\ge 2$. Here $\rho _k(G)=n-2=\varDelta (G)-1$ and $rc_k(G)=k(n-1)=k\varDelta (G)$, $k\ge 2$. Again let $G_1=K_2+2K_1$ and $G_2=C_4$. Then $rc_2(G_1)=2=2\varDelta (G_1)$ and $\rho _2(G_1)=0=\varDelta (G_1)-1$. Also, $rc_2(G_2)=4=2\varDelta (G_2)$ and $\rho _2(G_2)=1=\varDelta (G_2)-1$.

In this perspective, the following theorem gives an insight into the structure of a graph G when $\rho _k(G)=\varDelta (G)-1$ and $rc_k(G)=k\varDelta (G)$.

Theorem 8

Let G be a graph with $\varDelta (G)\ge 2$ and $\rho _k(G)=\varDelta (G)-1$. If $rc_k(G)=k\varDelta (G)$, then $G=H+K_1$, where $\rho _k(H)=\varDelta (H)$ and $rc_k(H)=k\varDelta (H)$.

Proof

Let $rc_k(G)=k\varDelta (G)=k\varDelta$ and $L\in \varLambda _{\rho _k}(G)$ which has islands $I_0, I_1,\ldots , I_{\rho _k}$ with $lc(I_{j+1})=rc(I_{j})+k$, $0\le j\le \rho _k-1$, such that $|I_0|\le |I_1|\le \ldots \le |I_{\rho _k}|$. Since $\rho _k(G)=\varDelta (G)-1$, by Lemma 3, $|I_j|=1$ and so $I_j=\{kj\}$ for $0\le j\le \varDelta -2$. Therefore $lc(I_{\varDelta -1})=k\varDelta -k$. Since $\rho _k(G)=\varDelta -1$ and $rc_k(G)=k\varDelta$, $rc(I_{\varDelta -1})=k\varDelta$ and $|I_{\varDelta -1}|\ge 3$. Hence $lc(I_{\varDelta -1})\ne rc(I_{\varDelta -1})$.

Now using Corollary 1, for every vertex u with $L(u)\in I_j$, for $0\le j\le \varDelta -2$, we have $d(u)=\varDelta$ and hence u is not adjacent to any vertex whose color is a non-coastal element of $I_{\varDelta -1}$. Let v be a vertex such that $k\varDelta -k<L(v)<k\varDelta$. Then v is isolated in G. Therefore any vertex, which is colored with an element of $I_{\varDelta -1}$ other than the coastal elements, can be suitably recolored to obtain an $rc_k$-coloring of G with fewer number of $(k-1)$-holes than $\rho _k(G)$, a contradiction, unless $I_{\varDelta -1}$ has only one element, say x, other than its coastal elements and $l_x=1$. Hence $G=H+K_1$, where $\rho _k(H)=\rho _k(G)+1=\varDelta (H)$ and $rc_k(H)=rc_k(G)=k\varDelta (G)=k\varDelta (H)$.$\square$

5.3 Relation among $(k-1)$-hole index, path covering number and island sequence

The following two results deal with $(k-1)$-hole index and path covering number. The last result of this subsection explores an interesting property of island sequence.

Theorem 9

For any graph $G\in {\mathcal {G}}_1\cup {\mathcal {G}}_2$, $\rho _k(G)=c(G^c)-1$, if $rc_k(G)\ge (n-1)(k-1)$, where n is the number of vertices of G.

Proof

Let $r=c(G^c)$.

Case I

Let $rc_k(G)=(n-1)(k-1)$. Then by Theorem 1, $G^c$ has a Hamiltonian path, say, $P:~x_0,~x_1,\ldots ,~x_{(n-1)}$. Define a radio k-coloring L on G by $L(x_i)=i(k-1)$, $0\le i\le n-1$. Then L is an $rc_k$-coloring on G and $\rho _k(G)=0=c(G^c)-1$.

Case II

Let $rc_k(G)>(n-1)(k-1)$. Then by Theorem 1, $r\ge 2$ and $rc_k(G)=n(k-1)+r-k$. Let ${\mathcal {P}}=\{P^{(1)},~P^{(2)},\ldots ,~P^{(r)}\}$ be a minimum path covering of $G^c$ and let the j-th vertex, $1\le j\le p_i$, of the $P^{(i)}$, $1\le i\le r$, be $x^i_j$, where $p_i$ is the number of vertices of $P^{(i)}$. We define a radio k-coloring L of G as $L(x^i_j)=(\varSigma ^{i-1}_{t=1}p_t-i+j)(k-1)+(i-1)k$. Then $span(L)=n(k-1)+r-k$, i.e., L is a $rc_k$-coloring on G. Also L has $(r-1)$ number of $(k-1)$-holes. Hence $\rho _k(G)\le r-1=c(G^c)-1$.

Let $\hat{L}\in \varLambda _{\rho _k}(G)$ and $I_0,~I_1,\ldots ,~I_{\rho _k(G)}$ be the islands of $\hat{L}$. Then the vertices in the set $A_i=\{u:~\hat{L}(u)\in I_i\}$ form a path $Q^{(i)}$ (say) in $G^c$, for $0\le i\le \rho _k(G)$. Thus $\{Q^{(0)},~Q^{(1)},\ldots ,~Q^{(\rho _k(G))}\}$ is a path covering of $G^c$. Hence $c(G^c)\le \rho _k(G)+1$.

Therefore $\rho _k(G)=c(G^c)-1$. This completes the proof.$\square$

Corollary 3

For any graph $G\in {\mathcal {G}}_1\cup {\mathcal {G}}_2$, if $c(G^c)\ge 2$ then $\rho _{k_1}(G)=\rho _{k_2}(G)$, for any $k_1,~k_2\ge 2$.

Proof

Proof follows directly from Theorems 1 and 9.$\square$

Corollary 4

For any graph $G\in {\mathcal {G}}_1\cup {\mathcal {G}}_2$, if $c(G^c)\ge 2$, then there exists a finite sequence of positive integers which is admitted as an island sequence by some $L\in \varLambda _{\rho _k}(G)$, for every $k\ge 2$.

Proof

Let $r=c(G^c)\ge 2$. Also let ${\mathcal {P}}=\{P^{(1)},~P^{(2)},\ldots ,~P^{(r)}\}$ be a minimum path covering of $G^c$ and $p_i$ be the number of vertices of $P^{(i)}$. Now by Theorem 1, for any $k\ge 2$, $rc_k(G)=n(k-1)+r-k$. Without loss of generality, we assume $p_1\le p_2\le \cdots \le p_r$. Let the j-th vertex, $1\le j\le p_i$, of $P^{(i)}$, $1\le i\le r$, be $x^i_j$. We define a radio k-coloring L of G as $L(x^i_j)=(\varSigma ^{i-1}_{t=1}p_t-i+j)(k-1)+(i-1)k$. Then L is a $rc_k$-coloring on G with $(r-1)=\rho _k(G)$ number of $(k-1)$-holes, by Theorem 9. So $L\in \varLambda _{\rho _k}(G)$. Now the colors of vertices in each path $P^{(i)}$ together form an island of L and each vertex receives distinct color. Therefore the island sequence of L is $(p_1,~p_2,\ldots ,~p_r)$. As $k\ge 2$ is arbitrary, we are done.$\square$