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Generation of Graceful Trees through Graceful CodesK. Balasubramanian$

N. Chandramowliswaran*

*Department of Mathematics, Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya(University), Kanchipuram.

Abstract

Graceful Code is a way to represent graceful graph in terms of sequence of non-negative integers. Given a graceful graph G on “q” edges, we can generate its graceful code in the form of (a1, a2, a3, …., aq-1, aq=0) to represent the graph. Similarly, we can easily draw the graph from the given graceful code. Graceful codes are classified into two categories, namely, α-valuable code and gracious code based on their properties. Graceful code provides an useful and efficient techniques to study and analyze graphs using computer. Here we discuss generation of infinitely many graceful codes, α-valuable codes and gracious codes for a given graceful code, α-valuable code and a gracious code.

Introduction

A simple graph G(V,E) on “p” vertices and “q”edges is said to be graceful if there exist an injection f: V→{0, 1, 2,….,q} such that the induced function g: E→{1, 2, 3, …, q} which is defined by g(u, v)=|f(u)-f(v)| for every edge (u, v), is a bijective function; then “f” is called graceful labelling of G.

Graceful Code

Let G be any graceful graph on “q” edges then (a1, a2, a3, …, aq -1, aq) is called a graceful code of G,if 0 ≤ ai ≤ q - i; 1 ≤ i ≤ q. Here ai is the lower end vertex of

the edge label “i”. It is important to note that aq is always zero

For every graceful graph G we can write its code. Conversely, for every given graceful code we can draw the corresponding graceful graph as follows.

Join edges:{(a1,1+a1),(a2,2+a2), …,(aq - 1,q-1+aq-1), (aq, q+aq)}

Example 1

Figure 1 shows a graceful graph on q = 7 edges with edge labeled from 1 to 7.

7

0

63

12

45

Code = (4, 2, 3, 0, 1, 0, 0)

α- valuable Code

A graceful code (a1, a2, a3,..., aq-1, aq) of a graceful graph G on “q” edges is called α - valuable code if

Here a1 is called the separator or critical valueof the α- valuable code.

a1 ≥ ai

Max{ai| 1≤ i ≤ q}< Min{i+ai| 1≤ i ≤ q}.

Proposition

(a1, a2, a3, …,aq-1,aq) represents α-valuable code if and only if

0 ≤ (a1 – aq - i + 1) / q – i) ≤ 1for all i , 1 ≤ i ≤ q - 1

Equivalently (a1, a2, a3, …,aq-1,aq) represents an α- valuable code if and only if (a1 - aq, a1 - aq -1, …, a1 - a3, a1 - a2, 0) represents a code of a Graceful Graph.

Properties of Graceful Codes

1.1 If (a1, a2, a3,…, aq - 1, aq) represents a code of a graceful graph G on “q” edges, then,(a2, a3,…, aq - 1, aq) represents a code of some graceful graph H on “q - 1” edges.

1.2 If (a1, a2, a3,…, aq-1, aq) is an α–valuable code on “q” edges and (q -1 - a1 > a1)then (q – 1 - a1, q – 2 - a2, …, 1- aq - 1, 0, a1, a2, a3,…, aq) is an α–valuablecode on “2q” edges.

1.3. If (a1, a2, a3,…, aq - 1, aq) is an α–valuable code on “q” edges and (a1> q -1 - a1) then,(a1, a2, a3,…, aq, q - 1- a1, q – 2 - a2, …, 1 – aq - 1, 0) is an α–valuablecode on “2q” edges.


1.4. If (a1, a2, a3,…, aq1 - 1, aq1) and (b1, b2, b3, …, bq2 - 1, bq2) represents α–valuable codes on “q1” and “q2” edges respectively and a1 ≥ b1 then,(a1, a2, a3,…, aq1 - 1, aq1, b1, b2, b3,…, bq2 - 1, bq2) represents an α–valuable code on “q1 + q2” edges.

1.5. Let (a1, a2, a3,…, aq -1, aq) represents a graceful code of a graph G on “q” edges then,(aq+ q, aq - 1+ q - 1,…, 2 + a2, 1 + a1, a1, a2, a3,…, aq - 1, aq) represents a α–valuable code on “2q” edges.


If (a1, a2, a3,…, aq - 1, aq) represents a graceful code of a graceful graph G on “q” edges then,(aq+ q, aq – 1 + q - 1,…,2 + a2, 1 + a1, x, a1, a2, a3…, aq - 1, aq), [0 ≤ x ≤ q] represents an α–valuable code “2q + 1” edges.

If (a1, a2, …, aq − 1, aq) represent a code of a graceful graph Gon q edges, Then,(q – aq, q – aq − 1, …, q – a2, q – a1, a1, a2, …, aq − 1, aq) represent a α-valuable code on “2⋅ q” edges.


Let X1, X2, X3, …, Xr represent “r” α–valuable codes on edges “qi”(1≤ i ≤ r) having separators “si” respectively.Then,

r-1 r-2 r-3( ∑ sj + Xr , ∑ sj + Xr-1 , ∑ sj + Xr-2 , … s1+ s2+ X3, s1+ X2, X1)j=1 j=1 j=1

ralways represent a α–valuable code on ∑ qj edges.

j = 1

Tree Generation Theorems

Let G be any simple graph on “n” vertices and “q” edges. Define a bipartite graph HG as follows:(vi, vj) ∈ E(G) <=> (vi, vj’) ∈ E(HG) and (vi’, vj) ∈ E(HG). Join any vk ∈ V(G) ⊆ V(HG), [1 ≤ k ≤ n] to vk’ ∈ V(HG). Here |V(HG)| = 2|V(G)| and |E(HG)| = 2 | E(G)| +1.Moreover if G has a code (a1, a2, a3,…, aq - 1, aq) then HG has an α–valuable code (aq+ q, aq-1+q - 1,…, 2+a2, 1 + a1, x, a1, a2, a3, …, aq - 1, aq) [0 ≤ x ≤ q]. If G happens to be a bipartite graph, then HG contains two copies of Gtogether with an edge connecting vk to vk’

Examples

Code = (0, 1, 0, 0)

G

HG

Code = (4, 3, 3, 1, 3, 0, 1, 0, 0)

Examples

G

Construction of HG

⇒

ai i + ai

E(HG)

i

ai i + ai q+1+iq+1-iE (G)q+1+ai q+1+i+ai

i i(aq+q, …,ai+i, …,1+a1, x, a1, …, ai, …, aq)

q+1+i+ai

+1+iq+1+ai

q+1-i q


TheoremIf (a1, a2, a3,…, aq -1, aq) represents a α–valuable code of some tree “T”Then,(aq+q, aq - 1+q-1, …,2+a2, 1+a1, a1, a2, a3,…, aq - 1, aq) represents a α–valuable code of a tree “S” on “2q” edges such that

E(S) = E(T) U E(T).

TheoremIf (a1, a2, a3,…, aq-1, aq) is an α–valuable code of a graceful graph G on “q” edges, then,(a1, a2, a3,…, aq-1, aq) represents a tree if and only if (a2, a3,…, aq -1, aq) represents a tree on “q - 1” edges.


If (a1, a2, …, aq-2, aq-1, aq ) represents a code of a graceful tree on ‘q’ edges, then

1. (qk - 1, ka1, (q –1) k –1, ka2, …, 2 k - 1, kaq - 1, 1k - 1, kaq) represent a tree code on “kq” edges (k ≥ 2).

2. (qk - 1, ka1+r, (q –1) k – 1, ka2+r, …, 2k - 1, kaq-1+r, 1k - 1, kaq+r, 0r) ;1 ≤ r ≤ k, k ≥ 2 represent a tree code on “kq+r” edges.

Corollary – 1If (a1, a2, …, aq - 2, aq - 1, aq ) represents a code of a graceful tree on ‘q’edges, then (q, 2a1, q - 1, 2a2, q - 2, 2a3, …, 2, 2aq - 1, 1, 2aq) represent a code of a graceful tree on “2q” edges and (q, 2a1+1, q - 1, 2a2+1, q - 2, 2a3+1, …, 2, 2aq - 1+1, 1, 2aq+1,0) represent a tree code on “2q+1” edges.

Corollary – 2If (a1, a2, …, aq-2, aq-1, aq ) represents a code of a graceful tree on ‘q’edges, then (q+1, 2a1, q, 2a2, q - 1, 2a3, …, 3, 2aq-1, 2, 2aq, 1, 0) represent a code of a graceful tree on “2q+2” edges and (q+1, 2a1+1, q, 2a2+1, q - 1,2a3+1, …, 3, 2aq - 1+1, 2, 2aq+1, 1, 0, 0) represent a tree code on “2q + 3” edges.

Tree Generation Theorems Using α- valuable tree codesTheorem 1If (a1, a2, …, aq - 1, aq) represent a α-valuable tree code on “q” edges, then,(aq+q, aq - 1+ q – 1, …, 2 + a2, 1+ a1, 1 + a1, a1, a1, a2, …, aq - 1, aq)represent a α-valuable tree code on “2q+2” edges.

Theorem 2Let (a1, a2, …, aq1 - 2, aq1 - 1, aq1) represent a α-valuable tree code on “q1”edges and (b1, b2, …, bq2 - 2, bq2 - 1, bq2) represent a tree code on “q2” edges. Then,(a1 + b1 , a1 + b2, …, a1 + bq2 - 2, a1+ bq2 - 1, a1+ bq2, a1, a2, …, aq1 - 2, aq1 - 1, aq1 ) represent a tree code on “q1 + q2” edges.

Tree Generation Theorems Using α- valuable tree codes2. (a1+ b1, a1+b2, …, a1+ bq2 - 2, a1+bq2 - 1, a1+bq2, a1 – aq1, a1 – aq1 - 1,

a1− aq1 - 2, …, a1 – a2, 0) represent a tree code on “q1+ q2” edges.

3. (q1– 1 – a1 + b1, q1– 1 – a1+ b2, q1– 1 – a1+ b3, …, q1– 1– a1+ bq2 - 2,q1– 1– a1+ bq2 - 1, q1– 1 – a1+ bq2, q1– 1– a1, q1 – 2 – a2, …, 2 − aq1 - 2, 1 − aq1 - 1, 0) represent a tree code on “q1+ q2” edges.

4. (q1– 1 – a1+ b1, q1– 1 – a1+ b2, q1– 1 – a1+ b3 , …, q1– 1 – a1+ bq2 - 2, q1– 1 – a1+ bq2 - 1, q1– 1 – a1+ bq2, q1– 1 – a1, q1– 2 – (a1 − aq1 - 1), q1 – 3 – (a1 − aq1 - 2), …, 1 – (a1 – a2), 0) represent a tree code on “q1+ q2” edges.

Tree Generation Theorems Using α- valuable tree codes5. (a1+ a2, a1+ a3, a1+ a4, …, a1+ aq1 - 2, a1+ aq1 - 1, a1+ aq1, a1, a2, …, aq1 - 2,

aq1-1, aq1) represent a tree code on “2q – 1” edges.

Corollary 1 Let X1, X2, X3, …, Xr represent “r” α–valuable tree codes on edges “qi”(1≤ i ≤ r) having separators “si” respectively.Then r- 1 r- 2 r- 3

( ∑ sj + Xr, ∑ sj + Xr - 1, ∑ sj + Xr - 2, …, s1+ s2+ X3, s1+ X2, X1) j=1 j=1 j=1

ralways represent a α–valuable tree code on ∑ qj edges.

j=1

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