Graphoidal Tree d - Cover

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1 International J.Math. Combin. Vol. (009), Graphoidal ree d - Cover S.SOMASUNDARAM (Department of Mathematics, Manonmaniam Sundaranar University, irunelveli 67 01, India) A.NAGARAJAN (Department of Mathematics, V.O. Chidambaram College, uticorin , India) G.MAHADEVAN (Department of Mathematics, Gandhigram Rural University, Gandhigram 64 30, India) gmaha003@yahoo.co.in Abstract: In [1] Acharya and Sampathkumar defined a graphoidal cover as a partition of edges into internally disjoint (not necessarily open) paths. If we consider only open paths in the above definition then we call it as a graphoidal path cover [3]. Generally, a Smarandache graphoidal tree (k, d)-cover of a graph G is a partition of edges of G into trees 1,,, l such that E( i) E( j) k and i d for integers 1 i, j l. Particularly, if k = 0, then such a tree is called a graphoidal tree d-cover of G. In [3] a graphoidal tree cover has been defined as a partition of edges into internally disjoint trees. Here we define a graphoidal tree d-cover as a partition of edges into internally disjoint trees in which each tree has a maximum degree bounded by d. he minimum cardinality of such d-covers is denoted by (G). Clearly a graphoidal tree -cover is a graphoidal cover. We find γ(d) (G) for some standard graphs. Key Words: graphoidal cover. Smarandache graphoidal tree (k, d)-cover, graphoidal tree d-cover, AMS(000): 05C70 1. Introduction hroughout this paper G stands for simple undirected graph with p vertices and q edges. For other notations and terminology we follow []. A Smarandache graphoidal tree (k, d)-cover of G is a partition of edges of G into trees 1,,, l such that E( i ) E( j ) k and i d for integers 1 i, j l. Particularly, if k = 0, then such a cover is called a graphoidal tree d-cover of G. A graphoidal tree d-cover (d ) F of G is a collection of non-trivial trees in G such that (i) Every vertex is an internal vertex of at most one tree; (ii) Every edge is in exactly one tree; (iii) For every tree F, () d. 1 Received March 6, 009. Accepted June 5, 009.

2 Graphoidal ree d - Cover 67 Let G denote the set of all graphoidal tree d-covers of G. Since E(G) is a graphoidal tree d-cover, we have G. Let (G) = min J. hen γ(d) (G) is called the graphoidal tree J G d-covering number of G. Any graphoidal tree d-cover of G for which J = (G) is called a minimum graphoidal tree d-cover. A graphoidal tree cover of G is a collection of non-trivial trees in G satisfying (i) and (ii). he minimum cardinality of graphoidal tree covers is denoted by γ (G). A graphoidal path cover (or acyclic graphoidal cover in [5]) is a collection of non-trivial path in G such that every vertex is an internal vertex of at most one path and every edge is in exactly one path. Clearly a graphoidal tree -cover is a graphoidal path cover and a graphoidal tree d-cover ( d ) is a graphoidal tree cover. Note that γ (G) (G) for all d. It is observe that (G) d Preliminaries heorem.1([4]) γ (K p ) = p. heorem.([4]) γ (K n,n ) = n 3. heorem.3([4]) If m n < m 3, then γ (K m,n ) = m+n 3. Further more, if n > m 3, then γ (K m,n ) = m. heorem.4([4]) γ (C m C n ) = 3 if m, n 3. heorem.5([4]) γ (G) p if δ(g) p. 3. Main results We first determine a lower bound for γ (d)(g). Define n d = min J G d n J, where G d is a collection of all graphoidal tree d-covers and n J is the number of vertices which are not internal vertices of any tree in J. heorem 3.1 For d, γ (d)(g) q (p n d )(d 1). Proof Let Ψ be a minimum graphoidal tree d-cover of G such that n vertices of G are not internal in any tree of Ψ. Let k be the number of trees in Ψ having more than one edge. For a tree in Ψ having more than one edge, fix a root vertex which is not a pendant vertex. Assign direction to the edges of the k trees in such a way that the root vertex has in degree zero and every other vertex has in degree 1. In Ψ, let l 1 be the number of vertices of out degree d and l the number of vertices of out degree less than or equal to d 1 (and > 0) in these k trees. Clearly l 1 + l is the number of internal vertices of trees in Ψ and so l 1 + l = p n. In each tree of Ψ there is at most one vertex of out degree d and so l 1 k. Hence we have

3 68 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN k + q (l 1 d + l (d 1)) = k + q (l 1 + l )(d 1)l 1 = k + q (p n Ψ )(d 1) l 1 q (p n d )(d 1). Corollary 3. (G) q p(d 1). Now we determine graphoidal tree d-covering number of a complete graph. heorem 3.3 For any integer p 4, (K p) = p(p d+1) if d < p ; p if d p. Proof Let d p. We know that γ(d) (K p) γ (K p ) = p by heorem.1. Case (i) Let p be even, say p = k. We write V (K p ) = {0, 1,,, k 1}. Consider the graphoidal tree cover J 1 = { 1,,, k }, where each i ( i = 1,,, k ) is a spanning tree with edge set defined by E( i ) = {(i 1, j) : j = i, i + 1,, i + k 1} {(k + i 1, s) : s j(modk), j = i + k, i + k + 1,, i + k }. Now J 1 = k = p. Note that ( i) = k d for i = 1,,, k and hence γ (d)(k p ) = p. Case (ii) Let p be odd, say p = k + 1. We write V (K p ) = {0, 1,,, k}. Consider the graphoidal tree cover J = { 1,,, k+1 } where each i ( i = 1,,, k ) is a tree with edge set defined by E( i ) = {(i 1, j) : j = i, i + 1,, i + k 1} {(k + i 1, s) : s j(modk + 1), j = i + k, i + k + 1,, i + k 1}. E( k+1 ) = {(k, j) : j = 0, 1,,, k 1}. Now J = k = p. Note that the degree of every internal vertex of i is either k or k + 1 and so ( i ) d, i = 1,,, k + 1. Hence (K p) = p if d p. Let d < p. By Corollary 3., (K p(p 1) p(p d + 1) p) q + p pd = + p pd =. Remove the edges from each i in J 1 ( or J ) when p is even (odd) so that every internal vertex is of degree d in the new tree i formed by this removal. he new trees so formed together with the removed edges form J 3.

4 Graphoidal ree d - Cover 69 If p is even, then J 3 is constructed from J 1 and J 3 = k + q k(d 1) = k + k(k 1) k(d 10 = k(k d + 1) = p(p d + 1). If p is odd, then J 3 is constructed from J and J 3 = k+1+q k(d 1) d = k+1+ k(k + 1) kd+k d = (k+1)(1+k d) = p(p d + 1). Hence (K p) = p(p+1 d). he following examples illustrate the above theorem. Examples 3.4 Consider K 6. ake d = 3 = p and V (K 6) = {,,,,, }. v3 Whence γ (3) (K 6) = 3. ake d = < p. Fig. 1 Fig. Whence γ () (K 6) = 6 (6 + 1 ) = 9. Consider K 7. ake d = 4 = p and V (K 7) = {,,,,,, v 6 }.

5 70 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN v 6 v 6 v 6 v 6 Fig.3 Whence, γ (4) = 4 = p. Now take d = 3 < p. v 6 v 6 v 6 v 6 Fig.4 herefore, γ (3) (K 7) = 7 ( ) = 7. We now turn to some cases of complete bipartite graph. heorem 3.5 If n, m d, then (K m,n) = p + q pd = mn (m + n)(d 1). Proof By theorem 3., (K m,n) p + q pd = mn (m + n)d + m + n. Consider G = K d,d. Let V (G) = X 1 Y 1, where X 1 = {x 1, x,, x d } and Y 1 = {y 1, y,, y d }. Clearly deg(x i ) = deg(y j ) = d, 1 i, j d. For 1 i d, we define i = {(x i, y j ) : 1 j d}, d+i = {(x i+d, y j ) : d + 1 j d} d+i = {(y i, x j ) : d + 1 j d} and 3d+i = {(y i+d, x j ) : 1 j d}. Clearly, J = { 1,,, 4d } is a graphoidal tree d-cover for G. Now consider K m,n, m, n d. Let V (K m,n ) = X Y, where X = {x 1, x,, x m } and Y = {y 1, y,, y n }. Now for 4d + 1 i 4d + m d = m + d, we define i = {(x i d, y j ) : 1 j d}. For m + d + 1 i m + n, we define i = {(y i m, x j ) : 1 j d}. hen J = { 1,,, 4d, 4d+1,, m+d, m+d+1,, m+n } {E(G) [E( i ) : 1 i m + n]} is a graphoidal tree d-cover for K m,n. Hence J = p + q pd and so (K m,n) p + q pd = mn (m + n)(d 1) for m, n d. he following example illustrates the above theorem. Example 3.6 Consider K 8,10 and take d = 4.

6 Graphoidal ree d - Cover 71 x 1 x x 3 x 4 y 1 y y 3 y 1 y y 3 y 1 y y 3 y 1 y y 3 y 5 y 6 y 7 y 8 x 7 x 8 y 1 y y 5 y 6 y 7 y 8 y 5 y 6 y 7 y 8 y5 y 6 y 7 y 8 x5 x 7 x 8 x 7 x 8 y 3 y 5 y 6 y 7 x 7 x 8 x 7 x 8 x 1 x x 3 x 4 x 1 x x 3 x 4 x 1 x x 3 x 4 y 8 y 9 y 10 y 9 y 9 y 9 y 9 y 10 y 10 y 10 y 10 x 1 x x 3 x 4 x 1 x x 3 x 4 x 1 x x 3 x 4 x 7 x 8 x 7 x 8 Fig.5 Whence, γ (4) = = 6. heorem 3.7 (K d 1,d 1) = p + q pd = d 1. Proof By heorem 3., (K d 1,d 1) p + q pd = d 1. For 1 i d 1, we define i = {(x i, y j ) : 1 j d} {(y i, x d+j ) : 1 j d 1} {(x d+i, y d+j ) : 1 j d 1}. Let d = {(x d, y j ) : 1 j d} {(y d, x d+j ) : 1 j d 1}. For d + 1 i d 1, we define i = {(y i, x j ) : 1 j d}. Clearly J = { 1,,, d 1 } is a graphoidal tree d-cover of G and so (G) d 1 = (d 1)(d 1 (d 1)) = q + p pd. he following example illustrates the above theorem. Example 3.8 Consider K 9,9 and d = 5.

7 7 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN y y 3 y 5 y 1 y 3 y 5 y 1 y y 5 y 1 y y 3 y 5 y 1 y y 3 x 1 x x 3 x 4 x 9 x 9 x 9 y 1 y y 3 x 8 x 8 x 8 x 7 x 7 x7 x 8 x 9 x 7 y 5 y 6 y 7 y 8 y 9 y 6 y 7 y 8 y 9 y 6 y 7 y 8 y 9 y 6 y 7 y 8 y 9 x 7 x 8 x 9 y 6 y 7 y 8 y 9 x 1 x x 3 x 4 x 1 x x 3 x 4 x 1 x x 3 x 4 x 1 x x 3 x 4 Fig.6 hereafter, γ (5) (K 9,9) = = 9. Lemma 3.9 (K 3r,3r) r, where d r and r > 1. Proof Let V (K 3r,3r ) = X Y, where X = {x 1, x,, x 3r } and Y = {y 1, y,, y 3r }. Case (i) r is even. For 1 s r, we define s = {(x s, y s+i ) : 0 i r 1} {(x s, y r+s )} {(x r+s, y r+s )} {(y r+s, x r+s )} {(x i, y r+s ) : 1 i r, i s} {(x r+s, y i ) : r + s i 3r, i r + s} {(x r+s, y i ) : 1 i s 1, s 1} and r+s = {(y s, x s+i ) : 1 i r} {(y s, x r+s )} {(y r+s, x r+s )} {(y i, x r+s ) : 1 i r, i s, r + 1 i 3r, i r + s} {(y r+s, x i ) : r + s + 1 i 3r, 1 i s, i r + s}. hen J 1 = { 1,,, r } is a graphoidal tree d-cover for K 3r,3r, ( i ) r and d r. So we have, ( K3r,3r) r. Case (ii) r is odd.

8 Graphoidal ree d - Cover 73 For 1 s r, we define s = {(x s, y s+i ) : 0 i r 1} {(y r+s, x i ) : r + 1 i 3r, i r + s} {(x r+s, y i ) : r + s i 3r} {(x r+s, y i ) : 1 i s 1, s 1} r+s = {(y s, x s+i ) : 1 i r} {(x r+s, y i ) : r + 1 i 3r; i = r + s} {(y r+s, x i ) : r + s + 1 i 3r, s r} {(y r+s, x i ) : 1 i s}. Clearly ( i ) r for each i. In this case also J = { 1,,, r } is a graphoidal tree d-cover for K 3r,3r and so (K 3r,3r) r when r is odd. he following example illustrates the above lemma for r =, 3. Consider K 6,6 and K 9,9. y1 x 1 y x x y 1 x 3 y y y 5 y 3 y 6 x 3 x 4 x 3 x x x 4 1 y y 6 y 3 y 1 y 5 y 3 y 6 y 1 y 5 x 1 x 4 x 1 x Fig.7 y 1 x 1 y 6 y x y 7 y 3 x 3 y 8 y y y y x 9 x 7 x8 y 7 y8 y 9 x y 1 x 7 y 3 y4 y y 6 5 x x x x x 7 6 y 8 y 9 y 1 x 3 y x 8 y 5 y 6 y 7 x 4 x5 x 7 x 9 x8 x 4 y 3 x 9 x 3 x 4 y 7 y8 y 9 x 4 x5 x 7 y 7 y 8 y5 y 9 x6 x7 x 8 y 6 y 7 y 8 y 9 x 8 x 9 x 1 x 1 x x 9 x 1 x x 3 Fig.8

9 74 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN heorem 3.10 (K n,n) = n 3 for d n 3 and n > 3. Proof By heorem., n 3 = γ (K n,n ) and γ (K n,n ) (K n,n), it follows that (K n,n) n 3 for any n. Hence the result is true for n 0(mod3). Let n 1(mod3) so that n = 3r + 1 for some r. Let J 1 = { 1,,, r } be a minimum graphoidal tree d-cover for K 3r,3r as in Lemma 3.9. For 1 i r, we define i = i {(x i, y 3r+1 )}, r+i = r+i {(y i, x 3r+1 )} and r+1 = {(x 3r+1, y r+i ) : 1 i r + 1} {(y 3r+1, x r+i ) : 1 i r}. Clearly J = { 1,,, r+1 } is a graphoidal tree d-cover for K 3r+1,3r+1, as ( i ) r + 1 = n 3 d for each i. Hence γ(d) (K n,n) = (K 3r+1,3r+1) r + 1 = n 3. Let n (mod3) and n = 3r+ for some r. Let J 3 be a minimum graphoidal tree d-cover for K 3r+1,3r+1 as in the previous case. Let J 3 = { 1,,, r+1 }. For 1 i r, we define i = i {(x i, y 3r+ )}, r+i = r+i {(y i, x 3r+ )}, r+1 = r+1, r+ = {(x 3r+, x r+i ) : 1 i r + } {(y 3r+, x r+i ) : 1 i r + 1}. Clearly, J 4 = { 1,,, r+} is a graphoidal tree d-cover for K 3r+,3r+, as ( i ) r+ = n 3 d for each i. Hence γ(d) (K n,n) = (K 3r+,3r+) r + = n 3. herefore, (K n,n) = n 3 for every n. Now we turn to the case of trees. heorem 3.11 Let G be a tree and let U = {v V (G) : deg(v) d > 0}. hen (G) = χ U (v)(deg(v) d) + 1, where d and χ U (v) is the characteristic function of U. v V (G) Proof he proof is by induction on the number of vertices m whose degrees are greater than d. If m = 0, then J = G is clearly a graphoidal tree d-cover. Hence the result is true in this case and (G) = 1. Let m > 0. Let u V (G) with deg G(u) = d + s ( s > 0 ). Now decompose G into s + 1 trees G 1, G,, G s, G s+1 such that deg Gi (u) = 1 for 1 i s, deg Gs+1 (u) = d. By induction hypothesis, (G i) = deg Gi (v)>d (deg Gi d) + 1 = k i, 1 i s + 1. Now J i is the minimum graphoidal tree d-cover of G i and J i = k i for 1 i s + 1. Let J = J 1 J J s+1. Clearly J is a graphoidal tree d-cover of G. By our choice of u, u is internal in only one tree of J. More over, deg (u) = d and deg Gi (v) = deg G (v) for v u and v V (G i ) for 1 i s + 1. herefore,

10 Graphoidal ree d - Cover 75 s+1 J = k i = = = = s+1 deg Gi (v)>d deg G(v)>d,v u v V (G) s+1 deg Gi (v)>d (deg Gi (v) d) + 1 (deg Gi (v) d) + s + 1 = deg G(v)>d,v u (deg G (v) d) + (deg G (u) d) + 1 = χ U (v)(deg G (v) d) + 1. (deg G (v) d) + s + 1 deg G(v)>d (deg G (v) d) + 1 For each v V (G) and deg G (v) > d there are at least deg G (v) d + 1 subtrees of G in any graphoidal tree d-cover of G and so (G) (deg G (v) d) + 1. Hence deg G(v)>d (G) = χ U (v)(deg G (v) d) + 1. v V (G) Corollary 3.1 Let G be a tree in which degree of every vertex is either greater than or equal to d or equal to one. hen (G) = m(d 1) p(d ) 1, where m is the number of vertices of degree 1 and d. Proof Since all the vertices of G other than pendant vertices have degree d we have, = v V (G) χ U (v)(deg G (v) d) + 1 = v V (G) χ U (v)(deg G (v) d) + md m + 1 = q dp + md m + 1 = p dp + md m + 1 (as q = p 1) = m(d 1) p(d ) 1. Recall that n d = min n J and n = min n J, where G d is the collection of all graphoidal J G d J G tree d-covers of G, G is the collection of all graphoidal tree covers of G and n J is the number of vertices which are not internal vertices of any tree in J. Clearly n d = n if d. Now we prove this for any d. Lemma 3.13 For any graph G, n d = n for any integer d. Proof Since every graphoidal tree d-cover is also a graphoidal tree cover for G, we have n n d. Let J = { 1,,, m } be any graphoidal tree cover of G. Let Ψ i be a minimum graphoidal tree d - cover of i (i = 1,,, m). Let Ψ = m Ψ i. Clearly Ψ is a graphoidal tree d-cover of G. Let n Ψ be the number of vertices which are not internal in any tree of Ψ. Clearly n Ψ = n J. herefore, n d n Ψ = n J for J G, where G is the collection of graphoidal tree covers of G and so n d n. Hence n = n d. We have the following result for graphoidal path cover. his theorem is proved by S.

11 76 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN Arumugam and J. Suresh Suseela in [5]. We prove this, by deriving a minimum graphoidal path cover from a graphoidal tree cover of G. heorem 3.14 γ () (G) = q p + n. Proof From heorem 3.1 it follows that γ () (G) q p + n. Let J be any graphoidal tree cover of G and J = { 1,,, k }. Let Ψ i be a minimum graphoidal tree d-cover of i (i = 1,,, k). Let m i be the number of vertices of degree 1 in i (i = 1,,, k). hen by heorem 3.1 it follows that γ () ( i) = m i 1 for all i = 1,,, k. Consider the graphoidal tree -cover Ψ J = k Ψ i of G. Now Notice that Ψ J = Ψ i = (m i 1) = = q p i + m i + m i. q i p i p i = (numbers of internal vertices and pendant vertices of i ) = p n J + m i. herefore, Ψ J = q p + n. Choose a graphoidal tree cover J of G such that n J = n. hen for the corresponding Ψ J we have Ψ J = q p + n = q p + n, as n = n by Lemma Corollary 3.15 If every vertex is an internal vertex of a graphoidal tree cover, then γ () (G) = q p. Proof Clearly n = 0 by definition. By Lemma 3.13, n = n. So we have n = 0. J. Suresh Suseela and S. Arumugam proved the following result in [5]. However, we prove the result using graphoidal tree cover. heorem 3.16 Let G be a unicyclic graph with r vertices of degree 1. Let C be the unique cycle of G and let m denote the number of vertices of degree greater than on C. hen γ () (G) = if m = 0, r + 1 m = 1, deg(v) 3 where v is the unique vertex of degree > on C, r oterwise.

12 Graphoidal ree d - Cover 77 Proof By Lemma 3.13 and heorem 3.14, we have γ () (G) = q p+n. We have q(g) = p(g) for unicyclic graph. So we have γ () (G) = n. If m = 0, then clearly γ() (G) =. Let m = 1 and let v be the unique vertex of degree > on C. Let e = vw be an edge on C. Clearly J = G e, e is a minimum graphoidal tree cover for G and so n r + 1. Since there is a vertex of C which is not internal in a tree of a graphoidal tree cover, we have n = r + 1. When m = 1, γ () (G) = r +1. Let m. Let v and w be vertices of degree greater than on C such that all vertices in a (v, w) - section of C other than v and w have degree. Let P denote this (v, w)-section. If P has length 1. hen P = (v, w). Clearly J = G P, P is a graphoidal tree cover of G. Also n = r and so γ () (G) = r when m. Hence we get the theorem. heorem 3.17 Let G be a graph such that γ (G) (G) = q p(d 1). δ(g) d + 1 (δ(g) > d ). hen Proof By heorem 3., (G) q p(d 1). Let J be a minimum graphoidal tree cover of G. Since δ > γ (G), every vertex is an internal vertex of a tree in a graphoidal tree cover J. Moreover, since δ d +δ (G) 1 the degree of each internal vertex of a tree in J is d. Let Ψ i be a minimum graphoidal tree d-cover of i ( i = 1,,, k ). Let m i be the number of vertices of degree 1 in i ( i = 1,,, k ). hen by Corollary 3.1, for i = 1,,, k we have γ () ( i) = p i (d ) + m i (d 1) 1. Consider the graphoidal tree d-cover Ψ = k Ψ i of G. Notice that Ψ = = = k Ψ i = (m i (d 1) p i (d ) 1) (m i (d 1) p i (d ) + q i p i ) [(m i p i )(d 1) + q i ] = (d 1) = (d 1) (m i p i ) + (m i p i ) + q. q i p i = (numbers of internal vertices and pendant vertices of i ) = p + m i.

13 78 S.SOMASUNDARAM, A.NAGARAJAN and G.MAHADEVAN herefore, Ψ = (d 1) + q. In other words, (G) q p(d 1). Hence, γ(d) (G) = q p(d 1) Corollary 3.18 Let G be a graph such that δ(g) = p + k where k 1. hen γ(d) (G) = q p(d 1) for d k + 1. Proof δ(g) d + 1 = p + k d + 1 p γ (G) by heorem.5. Applying heorem 3.17, (G) = q p(d 1). Corollary 3.19 Let G be an r-regular graph, where r > p. hen γ(d) (G) = q p(d 1) for d r + 1 p. Proof Here δ(g) = r and so the result follows from Corollary Corollary 3.0 (K m,n) = q p(d 1), where d m n 3 and 6 m n m 6. Proof Consider δ(g) d + 1 m m n 3 = m + n = 3m m + n m + n = γ (K m,n ). 3 Hence by Corollary 3.18, (K m,n) = q p(d 1). heorem 3.1 (C m C n ) = 3 for d 4 and γ () (C m C n ) = q p. Proof For d (G) = 4, (C m C n ) = γ (C m C n ) = 3 by heorem.14. Since δ(c m C n ) = 4 and γ (C m C n ) = 3, we have γ (C m C n ) = δ(g) d + 1 when d =. Applying heorem 3.17, γ () (C m C n ) = q p. References [1] Acharya B.D., Sampathkumar E., Graphoidal covers and graphoidal covering number of a graph, Indian J.Pure Appl. Math., 18(10), , October [] Harary F., Graph heory, Addison-Wesley, Reading, MA, [3] Somasundaram, S., Nagarajan., A., Graphoidal tree cover, Acta Ciencia Indica, Vol.XXIIIM. No.., (1997), [4] Suresh Suseela, J. Arumugam S., Acyclic graphoidal covers and path partitions in a graph, Discrete Math., 190 (1998),