On Eulerian Blict Graphs and Blitact Graphs

On Eulerian Blict Graphs and Blitact Graphs

V.R. Kulli

and M.S. Biradar

1

Department of Mathematics

Gulbarga University, Gulbarga, 585106, India [email protected]

2

Department of Mathematics

Govt. First Grade College, Basavakalyan, India.

(Received on: December 27, 2015)

ABSTRACT

In this paper, we establish a necessary and sufficient condition for blict graphs and blitact graphs of connected graphs to be eulerian.

AMS Mathematics Subject Classification: 05C Keywords: blict graph, blitact graph, eulerian.

1. INTRODUCTION

All definitions and notations not given in this paper may found in Kulli

. The graphs considered are finite, undirected without loops or multiple lines.

If B = {u

, u

, ... u

; r2} is a block of a graph G, then we say that point u

and block B are incident with each other, as are u

and B and so on. If two distinct blocks B

and B

are incident with a common cutpoint, then they are adjacent blocks. This idea was introduced by Kulli

. The blocks, cutpoints and lines of a graph are called its members.

The blict graph B

(G) of a graph G is the graph whose set of points is the union of the set of blocks, cutpoints and lines of G and in which two points are adjacent if the corresponding blocks and lines of G are adjacent or the corresponding members of G are incident. The blitact graph B

(G) of a graph G is the graph whose set of points is the union of the set of blocks, cutpoints and lines of G and in which two points are adjacent if the corresponding members of G are adjacent or incident. These concepts were introduced by Kulli and Biradar

. Many other graph valued functions in graph theory were studied, for example, in

.

The aim of this paper is to establish characterizations of graphs whose blict graphs and

blitact graphs are eulerian. Eulerian properties of some graph valued functions were studied,

The following result is useful to prove our results.

Theorem A [1, p.76]. A connected graph G is eulerian if and only if every point of G is of even degree.

The line graph L(G) of G is the graph whose point set corresponds to the lines of G such that two points of L(G) are adjacent if the corresponding lines of G are adjacent. This concept was studied, for example, in [31, 32, 33, 34, 35, 36, 37].

2. EULERIAN BLICT GRAPHS

The following will be useful in the proof our result.

Theorem 1 [3]. Let G be a connected (p, q) graph. Then B

(G) = L(G) K

, if and only if G is a block.

Lemma 2. If a point v of B

(G) corresponds to a cutpoint c of a graph G, then deg

B Gn v

 

where n

is the number of lines incident with a cutpoint c and n

is the number of blocks incident with a cutpoint c.

Lemma 3. If a point u of B

(G) corresponds to a line e of a graph G, then deg

B Gn u





where m

is the number of lines adjacent with a line e and m

is the number of cutpoints incident with a line e.

Lemma 4. If a point w of B

(G) corresponds to a block b of a graph G, then deg

B Gn w

 

where k

is the number of blocks adjacent with a block b and k

is the number of cutpoints incident with a block b.

Now we obtain a characterization of graphs whose blict graphs are eulerian.

Theorem 5. The blict graph B

(G) of a connected graph G is eulerian if and only if G satisfies the following conditions:

(1) G is separable,

(2) if e is a line of G, then the number of lines adjacent with e and the number of cutpoints incident with e are all either even or odd,

(3) if b is a block of G, then the number of blocks adjacent with b and the number of cutpoints incident with b are all either even or odd, and

(4) if v is a cutpoint of G, then the number of lines and blocks incident with v are all either even

or odd.

Proof: Suppose B

(G) is eulerian. Assume G is a nonseparable graph. Then G is a block. By Theorem 1, B

(G) = L(G) K

. Thus B

(G) is a disconnected graph and hence B

(G) is not eulerian which is a contradiction. Thus (1) holds.

Let e be a line of a connected graph G. By Lemma 3, deg

B Gn e





, where m

is the number of lines adjacent with e and m

is the number of cutpoints incident with e since

deg

B Gn e

is even, it implies that m

and m

are both either even or odd. Thus (2) holds.

Let b be a block of a connected separable graph G. By Lemma 4, deg

,

B Gn b

 

where k

is the number of blocks adjacent with b and k

is the number of cutpoints incident with b. Since deg

B Gn b

is even, it implies that k

and k

are both either even or odd. Thus (3) holds.

Let v be a cutpoint of a connected separable graph G. By Lemma 2, deg

,

B Gn v

 

where n

is the number of lines incident with v and n

is the number of blocks incident with v.

Since deg

B Gn v

is even, it implies that n

and n

are both either even or odd. Thus (4) holds.

Conversely suppose G satisfies conditions (1) to (4). Let v be a point of B

(G). Then v is either a line or a block or a cutpoint of G.

Suppose v is a line of G. By Lemma 3, deg

.

B Gn v





Then by condition (2), deg

B Gn v

is even.

Suppose v is a block of G. By Lemma 4, deg

.

B Gn v

 

Then by condition (3) deg

B Gn v

is even.

Suppose v is a cutpoint of G. By Lemma 2, deg

.

B Gn v

 

Then by condition (4), deg

B Gn v

is even.

Thus every point of B

(G) is even. Hence by Theorem A, B

(G) is eulerian.

3. EULERIAN BITACT GRAPHS

The following will be useful in the proof of our result.

Theorem 6 [3]. Let G be a connected (p, q) graph. Then B

(G) = L(G) K

if and only if G is a block.

Lemma 7. If a point u of B

(G) corresponds to a line of G, then deg

deg

.

m n

B G u



Lemma 8. If a point b of B

(G) corresponds to a block of G, then deg

deg

.

m n

B G b



Lemma 9. If a point v of B

(G) corresponds to a cutpoint c of G, then deg

deg

m n

B G v





where n is the number of cutpoints adjacent to c.

We establish a characterization of graphs whose blitact graphs are eulerian.

Theorem 10. The blitact graph B

(G) of a connected graph G is eulerian if and only if G satisfies the following conditions.

(1) G is separable,

(2) if e is a line of G, then the number of lines adjacent with e and the number of cutpoints incident with e are all either even or odd,

(3) if b is a block of G, then the number of blocks adjacent with b and the number of cutpoints incident with b are all either even or odd, and

(4) if v is a cutpoint of G, then the number of lines and blocks incident with v are all either even or odd and v is adjacent with an even number of cutpoints.

Proof: Suppose B

(G) is eulerian. Assume G is a nonseparable graph. Then G is a block. By Theorem 6, B

(G) = L(G) K

. Then B

(G) is a disconnected graph, which is a contradiction.

Thus (1) holds.

Let e be a line of G and e

be a point of B

(G) and it corresponds to e. By Lemma 7,

  1   1

deg deg .

m n

B G e



By Lemma 3, deg

,

BmG e





where m

is the number of lines adjacent with e and m

is the number of cutpoints incident with e. By Theorem A, deg

BmG e

is even, it implies that both m

and m

are either even or odd. Thus (2) holds.

Let b be a block of G and b

is a point of B

(G) and it corresponds to b. By Lemma 8,

  1   1

deg deg .

m n

B G b



By Lemma 4, deg

,

BmG b

 

where k

is the number of blocks adjacent with B and k

is the number of cutpoints incident with b. By Theorem A deg

B Gn b

is even, it implies that both k

and k

are either even or odd. Thus (3) holds.

Let c be a cutpoint of G. By Lemma 9, deg

,

B Gn c



where c

is corresponding point of c in B

(G) and n is the number of cutpoints adjacent with c. By Theorem A, deg

B Gn c

is even. By Theorem 5, deg

B Gn c

is even and hence n must be even. Thus (4) holds.

The proof of the converse part is similar to that of Theorem 5.

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