Path Double Covering Number of a Bicyclic Graphs

reproduction in any medium, provided the original work is properly cited. Abstract A path double cover of a graph G is a collection of paths in G such that every edge of G belongs to exactly two paths in. The minimum cardinality of a path double cover is called the path double covering number of G and is denoted by   PD G  .In this paper we determine the exact value of this parameter for several classes of graphs.


Introduction
A graph is a pair ,where V is the set of vertices and E is the set of edges.Here we consider only nontrivial, finite, connected undirected graph without loops or multiple edges.The order and size of G are denoted by p and q respectively.For graph theoretic terminology we refer to Harary [ 6 ].The concept of graphoidal cover was introduced by B.D Acharya and E.
Sampathkumar [1] and the concept of acyclic graphoidal cover was introduced by Arumugam and Suresh Suseela [4].The reader may refer [6] and [2] for the terms not defined here. is denoted by 1 P  .[3].Bondy [ 5 ] introduced the concept of path double cover of a graph.This was further studied by Hao Li [7].two cycles Cl and Cm where l,m  3,with a path of length i , i  1.Without loss of generality, we may assume is the chord.We denote this graph by Cm(i;l).

Main Results
In this paper we determine the value of PD  for U(l;m), D(l,m,i) and Cm(i;l) and for some classes of graphs.

Theorem 2.1
Let G be a bicyclic graph with n pendant vertices and G containing a   , U l m and j be the number of vertices greater than 2 on   , U l m other than 0 u .Let And let 12 j j j  Where 1 j and 2 j are the number of vertices of degree greater than 2 on l C and m C respectively.The path double cover is as follows , , , , , ,

T. Gayathri and S. Meena
Now assume that the result is true for all bicyclic graphs containing a   , U l m with j =1 and having less than n pendant vertices and The path double cover is as follows Let us assume that the result is true for all bicyclic graph containing a   , U l m with n-1 pendant vertices, j1= 0 or j2 = 0 and n > j and , , , , G G P P P P P  is a tree with n-3 pendant vertices.
  Sub case 3b: Let

T. Gayathri and S. Meena
The path double cover is as follows , , , , , , , , , , , (  ),  , ,  , , , , ,  (  ),  , ( ) ) Sub case 3c: j = 2 and n > j Without loss of generality assume that n = j+2 = 4 Similarly we can find the path double covering for the case j = 2 ,n > j and for the The path double cover is as follows , , , , Similarly we can find the path double covering for the case j =4 ,n > j So that   4 if G = , , j j 1and j 0 n+4 if j j 0 and j 1 j 1, 2, j j 0 j 2 or j 1and j 0 3if j j j 1 j or j 1, j 0 j 3, j 0,1, j 0 2 if j j 2, j 0 or 2 j j 1, j 0 Where j1, j2 and 3 j are the number of vertices greater than two on l C , m C and on the path respectively.The path double covering of G is as follows Sub Case 2a: Let called internal vertices of P and 1 v and r v are called external vertices of P. Two paths P and Q of a graph G are said to be internally disjoint if no vertex of G is an internal vertex of both P and Q .If in G then the walk obtained by concatenating P and Q at r v is denoted by PQ and the path  

Definition 1 . 1 [ 3 ]
: A path double cover (PDC) of a graph G is a collection of paths in G such that every edge of G belongs to exactly two paths in .The collection may not necessarily consist of distinct paths in G and hence it cannot be treated as a set in the standard sense.For any graph of all paths of length one each path appearing twice in the Path double covering number 1239 collection.Clearly is a path double of G and hence the set of all path double covers of G is non-empty.Arumugam and Meena [3] introduced the concept of path double covering number of a graph G. Definition 1.2[3]:The minimum cardinality of a path double cover of a graph G is called path double covering number of G and is denoted by   PD G  In [9] it has been observed that for any graph G   2 PD Gq   and equality holds if and only if G is isomorphic to 2 qK and the following results have been proved .Theorem1.3[3]:Let be any path double cover of a graph G. where i= maxiP the maximum being taken over all path double covers of G Theorem 1.5 [3]: Let G be a graph with 1   ,if there exists a path double cover such that every non pendant vertex of G is an internal vertex of d(v)paths in then is minimum path double cover and if T is homeomorphic to a star.Definition 1.8 [8]: A connected (p, p+1) graph G is called a bicyclic graph.T. Gayathri and S. MeenaDefinition 1.9 [8]: A onepoint union of two cycles is a simple graph obtained from two cycles, say Cl and Cm where l,m  3, by identifying one and the same vertex from both cycles.Without loss of generality, we may assume the l-cycle to A long dumbbell graph is a simple graph obtained by joining

.
We denote this graph by D(l,m,i) Definition 1.11 [8]: A cycle with a long chord is a simple graph obtained from an m-cycle, m 4  , by adding a chord of length l where l  1.Let the m-cycle be Without loss of generality, we may assume the chord joins 0 u with i u , where 22 im

1 j and 2 j
are the number of vertices of degree greater than 2 on the path double covering number of G is

j
 and j =1 We prove this by induction on n.When j =1 and n =1G is isomorphic to the graph consisting of   , U l m together with a path

5 v 5 v
Let G be any bicyclic graph containing a   , U l m with n pendant vertices and j =1Let w be any pendant vertex of G.Choose a vertex v such thatdeg  and d(w,v) is maximum.Let Q denote the (w,v) path, Since deg  there exist pendant vertices w1,w2 such that(w,w1) path Q1 and(w,w2) path Q2 both contain Q.Now let P1and P2 denote the (w1,v) section of Q1and (w2,v) section of Q2 Clearly v is the only vertex of degree greater than two on P. Either j1=0 or j2=0 and n > j and 2 j  We prove by induction on n Let j = 2, n = 3 (since 2 j  and n > j ) Path double covering number 1243 Without loss of generality assume that k u on l C is incident with the the path has a pendant vertex w1 and i u on l C incident with the path has pendant vertices w2 and w3 respectively.(i< k)

2 jw
 and Now let G be a bicyclic graph containing a   , U l m with n pendant vertices and j1= 0 or j2 = 0 and n > j and 2 j As before the mentioned paths P1to P5 covers the bicyclic as well as

u
Let j =2 and n = j Since j =2, clearly j1=1 and j2=1 is incident with the paths has pendant vertices 1 w and 2 w respectively.

C
The path double cover is as follows

Case 4 :
Let j = 4, n = j = 4 Let , ik uu are the two vertices on l C are of degree greater than two incident with the pendant vertices 1

Case 5 : 2 Let
Let j > 5, n > jHere the number of pendant vertices is greater than 5, we can find the path double covering such that all the vetices are internal vertices except the pendant vertices by using above proof technique.G be a bicyclic graph with n pendant vertices and G containing a   ,, D l m i is the unique bicycle in G and let j be the number of vertices of degree greater than

2 jCase 1 :
are the number of vertices of degree greater than 2 on l C and m C and j3 is the number of vertices of degree greater than 2 on the path respectively.

1247
We prove by induction on nLet n = 2 and let 00 , vw are pendant vertices.The path double covering of G is as

Let j = n = 1 G
Assume that the result is true for all bicyclic graph with n -1 pendant vertices containing a  ,, D l m iLet G be a graph bicyclic graph with n pendant vertices containing a   is isomorphic to the graph consisting of  