Labeling, Covering and Decomposing of Graphs Smarandache s Notion in Graph Theory

Transcription

1 International J.Math. Combin. Vol. (010), Labeling, Covering and Decomposing of Graphs Smarandache s Notion in Graph Theory Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing, ,.R.China) ( Beijing Institute of Civil Engineering and Architecture, Beijing, ,.R.China) maolinfan@16.com Abstract: This paper surveys the applications of Smarandache s notion to graph theory appeared in International J.Math.Combin. from Vol.1,008 to Vol.,009. In fact, many problems discussed in these papers are generalized in this paper. Topics covered in this paper include: (1)What is a Smarandache System? ()Vertex-Edge Labeled Graphs with Applications: (i)smarandachely k-constrained labeling of a graph; (ii)smarandachely super m-mean graph; (iii)smarandachely uniform k-graph; (iv)smarandachely total coloring of a graph; ()Covering and Decomposing of a Graph: (i)smarandache path k-cover of a graph; (ii)smarandache graphoidal tree d-cover of a graph; (4)Furthermore. Key Words: Smarandache system, labeled graph, Smarandachely k-constrained labeling, Smarandachely k-constrained labelingsmarandachely super m-mean graph, Smarandachely uniform k-graph, Smarandachely total coloring of a graph, Smarandache path k-cover of a graph, Smarandache graphoidal tree d-cover of a graph. AMS(000): 05C1, 05C70, 05C78 1. What is a Smarandache System? A Smarandache System first appeared in [1] is defined in the following. Definition 1.1([1]) A rule in a mathematical system (Σ; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Σ, i.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache system (Σ; R) is a mathematical system which has at least one Smarandachely denied rule in R. Definition 1.([]) For an integer m, let (Σ 1 ; R 1 ), (Σ ; R ),, (Σ m ; R m ) be m mathematical systems different two by two. A Smarandache multi-space is a pair ( Σ; R) with Σ = m m Σ i, and R = R i. i=1 1 Received June 1, 010. Accepted September 18, 010. Reported at Beijing Jiaotong University, November 18, 009. i=1

2 Labeling, Covering and Decomposing of Graphs 109 Definition 1.([]) An axiom is said to be Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969). Example 1.1 Let us consider an Euclidean plane R and three non-collinear points A, B and C. Define s-points as all usual Euclidean points on R and s-lines any Euclidean line that passes through one and only one of points A, B and C, such as those shown in Fig.1.1. (i) The axiom (A5) that through a point exterior to a given line there is only one parallel passing through it is now replaced by two statements: one parallel, and no parallel. Let L be an s-line passes through C and is parallel in the Euclidean sense to AB. Notice that through any s-point not lying on AB there is one s-line parallel to L and through any other s-point lying on AB there is no s-lines parallel to L such as those shown in Fig.1(a). (ii) The axiom that through any two distinct points there exist one line passing through them is now replaced by; one s-line, and no s-line. Notice that through any two distinct s- points D, E collinear with one of A, B and C, there is one s-line passing through them and through any two distinct s-points F, G lying on AB or non-collinear with one of A, B and C, there is no s-line passing through them such as those shown in Fig.1(b). L l 1 D C l 1 D C E A E B A F G B l (a) (b) Fig.1 Definition 1.4 A combinatorial system C G is a union of mathematical systems (Σ 1 ; R 1 ),(Σ ; R ),, (Σ m ; R m ) for an integer m, i.e., C G = ( m Σ i ; i=1 m R i ) i=1 with an underlying connected graph structure G, where V (G) = {Σ 1, Σ,, Σ m }, E(G) = { (Σ i, Σ j ) Σ i Σj, 1 i, j m}.. Vertex-Edge Labeled Graphs with Applications.1 Application to rincipal Fiber Bundles Definition.1 A labeling on a graph G = (V, E) is a mapping θ L : V E L for a label set L, denoted by G L.

3 110 Linfan Mao If θ L : E or θ L : V, then G L is called a vertex labeled graph or an edge labeled graph, denoted by G V or G E, respectively. Otherwise, it is called a vertex-edge labeled graph. Example: Fig. p M Definition.([4]) For a given integer sequence 0 < n 1 < n < < n m, m 1, a combinatorial manifold M is a Hausdorff space such that for any point p M, there is a local chart (U p, ϕ p ) of p, i.e., an open neighborhood U p of p in M and a homoeomorphism ϕ p : U p R(n 1 (p), n (p),, n s(p) (p)), a combinatorial fan-space with {n 1 (p), n (p),, n s(p) (p)} {n 1, n,, n m }, and {n 1 (p), n (p),, n s(p) (p)} = {n 1, n,, n m }, denoted by M(n 1, n,, n m ) or M on the context and an atlas on M(n 1, n,, n m ). Ã = {(U p, ϕ p ) p M(n 1, n,, n m ))} A combinatorial manifold M is finite if it is just combined by finite manifolds with an underlying combinatorial structure G without one manifold contained in the union of others. Certainly, a finitely combinatorial manifold is indeed a combinatorial manifold. Examples of combinatorial manifolds can be seen in Fig.. B 1 M T T B 1 B 1 (a) (b) Fig. Let M(n 1, n,, n m ) be a finitely combinatorial manifold and d, d 1 an integer. We construct a vertex-edge labeled graph G d [ M(n 1, n,, n m )] by V (G d [ M(n 1, n,, n m )]) = V 1 V, where V 1 = {n i manifolds M ni in M(n 1,, n m ) 1 i m} and V = {isolated intersection points O M n i,m n j ofm ni, M nj in M(n 1, n,, n m ) for 1 i, j m}. Label n i for each

4 Labeling, Covering and Decomposing of Graphs 111 n i -manifold in V 1 and 0 for each vertex in V and E(G d [ M(n 1, n,, n m )]) = E 1 E, where E 1 = {(M ni, M nj ) labeled with dim(m ni M nj ) dim(m ni M nj ) d, 1 i, j m} and E = {(O M n i,m n j, M ni ), (O M n i,m n j, M nj ) labeled with 0 M ni tangent M nj at the point O M n i,m n j for 1 i, j m}. Now denote by H(n 1, n,, n m ) all finitely combinatorial manifolds M(n 1, n,, n m ) and G[0, n m ] all vertex-edge labeled graphs G L with θ L : V (G L ) E(G L ) {0, 1,, n m } with conditions following hold. (1)Each induced subgraph by vertices labeled with 1 in G is a union of complete graphs and vertices labeled with 0 can only be adjacent to vertices labeled with 1. ()For each edge e = (u, v) E(G), τ (e) min{τ 1 (u), τ 1 (v)}. Then we know a relation between sets H(n 1, n,, n m ) and G([0, n m ], [0, n m ]) following. Theorem.1([1]) Let 1 n 1 < n < < n m, m 1 be a given integer sequence. Then every finitely combinatorial manifold M H(n 1, n,, n m ) defines a vertex-edge labeled graph G([0, n m ]) G[0, n m ]. Conversely, every vertex-edge labeled graph G([0, n m ]) G[0, n m ] defines a finitely combinatorial manifold M H(n 1, n,, n m ) with a 1 1 mapping θ : G([0, n m ]) M such that θ(u) is a θ(u)-manifold in M, τ 1 (u) = dimθ(u) and τ (v, w) = dim(θ(v) θ(w)) for u V (G([0, n m ])) and (v, w) E(G([0, n m ])). Definition.([4]) A principal fiber bundle consists of a manifold action by a Lie group G, which is a manifold with group operation G G G given by (g, h) g h being C mapping, a projection π : M, a base pseudo-manifold M, denoted by (, M, G ), seeing Fig.4 (where V = π 1 (U)) such that conditions (1), () and () following hold. (1) there is a right freely action of G on,, i.e., for g G, there is a diffeomorphism R g : with R g (p) = pg for p such that p(g 1 g ) = (pg 1 )g for p, g 1, g G and pe = p for some p, e G if and only if e is the identity element of G. () the map π : M is onto with π 1 (π(p)) = {pg g G }. () for x M there is an open set U with x U and a diffeomorphism T U : π 1 (U) U G of the form T U (p) = (π(p), s U (p)), where s U : π 1 (U) G has the property s U (pg) = s U (p)g for g G, p π 1 (U).

5 11 Linfan Mao V V V T U ¹ U G M x U π 1 Fig.4 Question For a family of k principal fiber bundles 1 (M 1, G 1 ), (M, G ),, k (M k, G k ) over manifolds M 1, M,, M k, how can we construct principal fiber bundles on a smoothly combinatorial manifold consisting of M 1, M,, M k underlying a connected graph G? The answer is YES. The technique is by voltage assignment on labeled graphs defined as follows. Definition.4([4]) A voltage labeled graph on a vertex-edge labeled graph G L is a -tuple (G L ; α) with a voltage assignments α : E(G L ) Γ such that with its labeled lifting G Lα defined by α(u, v) = α 1 (v, u), (u, v) E(G L ), V (G Lα ) = V (G L ) Γ, (u, g) V (G L ) Γ abbreviated to u g ; E(G L α) = { (u g, v g h ) for (u, v) E(G L ) with α(u, v) = h } with labels Θ L : G Lα L following: Θ L (u g ) = θ L (u), and Θ L (u g, v g h ) = θ L (u, v) for u, v V (G L ), (u, v) E(G L ) with α(u, v) = h and g, h Γ. For a voltage labeled graph (G L, α) with its lifting G L α, a natural projection π : G Lα G L is defined by π(u g ) = u and π(u g, v g h ) = (u, v) for u, v V (G L ) and (u, v) E(G L ) with α(u, v) = h. Whence, (G Lα, π) is a covering space of the labeled graph G L. A voltage labeled graph with its labeled lifting are shown in Fig.4.4, in where, G L = C L and Γ = Z. 4 1 (G L, α) 5 Fig Now we show how to construct principal fiber bundles over a combinatorial manifold M. G Lα 5

6 Labeling, Covering and Decomposing of Graphs 11 Construction.1 For a family of principal fiber bundles over manifolds M 1, M,, M l, such as those shown in Fig.6, H 1 H H l M1 M Ml Π M1 M 1 Π M M Π Ml M l Fig.6 where H i is a Lie group acting on Mi for 1 i l satisfying conditions FB1-FB, let M be a differentiably combinatorial manifold consisting of M i, 1 i l and (G L [ M], α) a voltage graph with a voltage assignment α : G L [ M] G over a finite group G, which naturally induced a projection π : G L [ ] G L [ M]. For M V (G L [ M]), if π( M ) = M, place M on each lifting vertex M Lα in the fiber π 1 (M) of G Lα [ M], such as those shown in Fig.7. M M M } {{ } π 1 (M) M Fig.7 Let Π = ππ M π 1 for M V (G L [ M]). Then = M is a smoothly combinatorial manifold and L G = M V (G L [ M]) combinatorial fiber bundle is denoted by Lα ( M, L G ). M V (G L [ M]) H M a Lie multi-group by definition. Such a constructed For example, let G = Z and G L [ M] = C. A voltage assignment α : G L [ M] Z and its induced combinatorial fiber bundle are shown in Fig.8. v 0 w 1 M M u 1 w 0 v 1 π u w v M M Π M M 1 M u 0 M1 M1 Fig.8

7 114 Linfan Mao Then we know the existence result following. Theorem.([4]) A combinatorial fiber bundle α ( M, L G ) is a principal fiber bundle if and only if for (M, M ) E(G L [ M]) and ( M, M ) = (M, M ) Lα E(G L [ ]), Π M M M = Π M M M.. Smarandachely k-constrained labeling of a graph In references [5]-[6], the Smarandachely k-constrained labeling on some graph families are discussed. Definition.5 A Smarandachely k-constrained labeling of a graph G(V, E) is a bijective mapping f : V E {1,,.., V + E } with the additional conditions that f(u) f(v) k whenever uv E, f(u) f(uv) k and f(uv) f(vw) k whenever u w, for an integer k. A graph G which admits a such labeling is called a Smarandachely k-constrained total graph, abbreviated as k CT G. An example for k = 5: Fig.9: A 5-constrained labeling of a path 7. Definition.6 The minimum positive integer n such that the graph G K n is a k CTG is called k-constrained number of the graph G and denoted by t k (G), the corresponding labeling is called a minimum k-constrained total labeling of G. roblem.1 Determine t k (G) for k Z + and a graph G. Update Results for roblem.1 obtained in [5]-[6]: Case 1. k = 1 In fact, t 1 (G) = 0 for any graph G since any bijective mapping f : V E {1,,.., V + E } satisfies that f(u) f(v) 1 whenever uv E, f(u) f(uv) 1 and f(uv) f(vw) 1 whenever u w. Case. k = 0 if n =, (1) t ( n )= 1 if n =, 0 else. roof Let V ( n ) = {v 1, v,..., v n } and E( n ) = {v i v i+1 1 i n 1}. Consider a total labeling f : V E {1,,,..., n 1} defined as f(v 1 ) = n ; f(v ) = n 1; f(v 1 v ) = ; f(v v ) = 4; and f(v k ) = k 5, f(v k v k+1 ) = k, for all k. This function f serves as a Smarandachely -constrained labeling for n, for n 4. Further, the cases n = and n =

8 Labeling, Covering and Decomposing of Graphs 115 are easy to prove n- n- n-1 1 n-7 n-5 () t (C n ) = 0 if n 4 and t (C ) =. Fig.10 roof If n 4, then the result follows immediately by joining end vertices of n by an edge v 1 v n, and, extending the total labeling f of the path as in the proof of the Theorem.4 above to include f(v 1 v ) = n. Consider the case n =. If the integers a and a + 1 are used as labels, then one of them is assigned for a vertex and other is to the edge not incident with that vertex. But then, a + can not be used to label the vertex or an edge in C. Therefore, for each three consecutive integers we should leave at least one integer to label C. Hence the span of any Smarandachely -constrained labeling of C should be at least 8. So t (C ). Now from the Figure it is clear that t (C ). Thus t (C ) =. () t (K n ) = 0 if n 4. (4) t (W 1,n ) = 0 if n. if n = 1 and m = 1, (6) t (K m,n )= 1 if n = 1 and m, 0 else. Case. k (1) t k (K 1,n )= k 6, if n =, n(k ), otherwise. if k.n. roof For any Smarandachely k-constrained labeling f of a star K 1,n, the span of f, after labeling an edge by the least positive integer a is at least a+nk. Further, the span is minimum only if a = 1. Thus, as there are only n + 1 vertices and n edges, for any minimum total labeling we require at least 1 + nk (n + 1) = n(k ) isolated vertices if n 4 and at least 1 + nk n = n(k )+1 if n =. In fact, for the case n =, as the central vertex is incident with each edge and edges are mutually adjacent, by a minimum k-constrained total labeling, the edges as well the central vertex can be labeled only by the set {1, 1 + k, 1 + k, 1 + k}.

9 116 Linfan Mao Suppose the label 1 is assigned for the central vertex, then to label the end vertex adjacent to edge labeled 1 + k is at least (1 + k) + 1 (since it is adjacent to 1, it can not be less than 1 + k). Thus at most two vertices can only be labeled by the integers between 1 and 1 + k. Similar argument holds for the other cases also. Therefore, t(k 1,n ) n(k ) for n 4 and t(k 1,n ) n(k ) + 1 for n =. To prove the reverse inequality, we define a k-constrained total labeling for all k, as follows: (1) When n =, the labeling is shown in the Fig.11 below k 1 + k 1 + k k Fig.11 () When n 4, define a total labeling f as f(v 0 v j ) = 1 + (j 1)k for all j, 1 j n. f(v 0 ) = 1 + nk, f(v 1 ) = + (n )k, f(v ) = + (n )k,and for i (n 1), f(v i ) +, if f(v i ) 0(mod k), f(v i+1 )= f(v i ) + 1, otherwise. and the rest all unassigned integers between 1 and 1 + nk to the n(k ) isolated vertices, where v 0 is the central vertex and v 1, v, v,..., v n are the end vertices. The function so defined is a Smarandachely k-constrained labeling of K 1,n K n(k ), for all n 4. () Let n be a path on n vertices and k 0 = n 1. Then 0 if k k 0, t k ( n )= (k k 0 ) 1 if k > k 0 and n 0(mod ), (k k 0 ) if k > k 0 and n 1 or (mod ). () Let C n be a cycle on n vertices and k 0 = n 1. Then 0 if k k 0, t k (C n ) = (k k 0 ) if k > k 0 and n 0 (mod ), (k k 0 ) if k > k 0 and n 1 or (mod ).

10 Labeling, Covering and Decomposing of Graphs 117. Smarandachely Super m-mean Graph The conception of Smarandachely edge m-labeling on a graph was introduced in [7]. Definition.7 Let G be a graph and f : V (G) {1,,,, V + E(G) } be an injection. For each edge e = uv and an integer m, the induced Smarandachely edge m-labeling f S is defined by f(u) + f(v) fs(e) =. m Then f is called a Smarandachely super m-mean labeling if f(v (G)) {f (e) : e E(G)} = {1,,,, V + E(G) }. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph. articularly, if m =, we know that f (e) = f(u)+f(v) if f(u) + f(v) is even; f(u)+f(v)+1 if f(u) + f(v) is odd. Example: A Smarandache super -mean graph Fig.1 roblem. Find integers m and graphs G such that G is a Smarandachely super m-mean graph. Update Results for roblem. Obtained in [7]: Now all results is on the case of Smarandache super -mean graphs. (1) A H-graph of a path n is the graph obtained from two copies of n with vertices v 1, v,...,v n and u 1, u,..., u n by joining the vertices v n+1 and u n+1 if n is odd and the vertices v n +1 and u n if n is even. Then A H-graph G is a Smarandache super -mean graph. () The corona of a graph G on p vertices v 1, v,..., v p is the graph obtained from G by adding p new vertices u 1, u,..., u p and the new edges u i v i for 1 i p, denoted by G K 1. If a H-graph G is a Smarandache super -mean graph, then G K 1 is a Smarandache super -mean graph.

11 118 Linfan Mao () For a graph G, the -corona of G is the graph obtained from G by identifying the center vertex of the star S at each vertex of G, denoted by G S. If a H-graph G is a Smarandache super -mean graph, then G S is a Smarandache super -mean graph. (4) Cycle C n is a Smarandache super -mean graph for n. (5) Corona of a cycle C n is a Smarandache super -mean graph for n. (6) A cyclic snake mc n is the graph obtained from m copies of C n by identifying the vertex v (k+)j in the j th copy at a vertex v 1j+1 in the (j + 1) th copy if n = k + 1 and identifying the vertex v (k+1)j in the j th copy at a vertex v 1j+1 in the (j + 1) th copy if n = k. The graph mc n -snake, m 1, n and n 4 has a Smarandache super -mean labeling. G. (7) A n (G) is a graph obtained from G by identifying an end vertex of n at a vertex of If G is a Smarandache super -mean graph then n (G) is also a Smarandache super -mean graph. (8) C m n for n 1, m =, 5 are Smarandache super -mean graphs. roblem. For what values of m (except,5) the graph C m n is a Smarandache super -mean graph?.4 Smarandachely Uniform k-graphs The conception of Smarandachely Uniform k-graph was introduced in the reference [8]. Definition.7 For an non-empty subset M of vertices in a graph G = (V, E), each vertex u in G is associated with the set fm o (u) = {d(u, v) : v M, u v}, called its open M-distancepattern. A graph G is called a Smarandachely uniform k-graph if there exist subsets M 1, M,, M k for an integer k 1 such that f o M i (u) = f o M j (u) and f o M i (u) = f o M j (v) for 1 i, j k and u, v V (G). Such subsets M 1, M,, M k are called a k-family of open distance-pattern uniform (odpu-) set of G and the minimum cardinality of odpu-sets in G, if they exist, is called the Smarandachely odpu-number of G, denoted by od S k (G). Usually, a Smarandachely uniform 1-graph G is called an open distance-pattern uniform (odpu-) graph. In this case, its odpu-number od S k (G) of G is abbreviated to od(g). roblem.4 Determine which graph G is Smarandachely uniform k-graph for an integer k 1. Update Results for roblem.4 Obtained in [8]: (1) A connected graph G is an odpu-graph if and only if the center Z(G) of G is an odpu-set. () Every self-centered graph is an odpu-graph.

12 Labeling, Covering and Decomposing of Graphs 119 () A tree T has an odpu-set M if and only if T is isomorphic to. (4) If G is a unicyclic odpu-graph, then G is isomorphic to a cycle. (5) A block graph G is an odpu-graph if and only if G is complete. (6) A graph with radius 1 and diameter is an odpu-graph if and only if there exists a subset M V (G) with M such that the induced subgraph M is complete, V M is not complete and any vertex in V M is adjacent to all the vertices of M. roblem.5 Determine the Smarandachely odpu-number od S k (G) of G for an integer k 1. Update Results for roblem.5 obtained in [8]: (1) For every positive integer k 1,, there exists a graph G with odpu-number k. () If a graph G has odpu-number 4, then r(g) =. () The number 5 cannot be the odpu-number of a bipartite graph. (4) Let G be a bipartite odpu-graph. Then od(g) = if and only if G is isomorphic to. (5) od(c k+1 ) = k. (6) od(k n ) = for all n..5 Smarandachely Total Coloring of a graph The conception of Smarandachely total k-coloring of a graph following is introduced by Zhongfu Zhang et al. in [9]. Definition.8 Let f be a total k coloring on G. Its total-color neighbor of a vertex u of G is denoted by C f (x) = {f(x) x T N (u)}. For any adjacent vertices x and y of V (G), if C f (x) C f (y), say f a k AVSDT-coloring of G (the abbreviation of adjacent-vertex-stronglydistinguishing total coloring of G). The AVSDT-coloring number of G, denoted by χ ast (G) is the minimal number of colors required for an AVSDT-coloring of G Definition.9 A Smarandachely total k-coloring of a graph G is an AVSDT-coloring with C f (x)\c f (y) k and C f (y)\c f (x) k. The minimum Smarandachely total k-coloring number of a graph G is denoted by χ k ast (G). Obviously, χ ast (G) = χ 1 ast(g) and by definition. χ k+1 ast (G) χ k ast(g) χ k 1 ast (G) χ 1 ast(g) roblem.6 Determine χ k ast(g) for a graph G. Update Results for roblem.6 obtained in [9]: χ 1 ast (S m + W n ) = m + n + if min{m, n} 5.

13 10 Linfan Mao It should be noted that the number χ k ast (G) of graph families following are determined for integers k 1 by Zhongfu Zhang et al. in references [10]-[15]. (1) -regular Halin graphs; () n, C n, K 1,n and double fan graphs for integers n 1; () m + n for integers m, n 1; (4) m n for integers m, n 1; (5) Generalized etersen G(n, k); (6) k-cube graphs.. Covering and Decomposing of a Graph Definition.1 Let be a graphical property. A Smarandache graphoidal (k, d)-cover of a graph G is a partition of edges of G into subgraphs G 1, G,, G l such that E(G i ) E(G j ) k and (G i ) d for integers 1 i, j l. The minimum cardinality of Smarandache graphoidal (k, d)-cover of a graph G is denoted by Π (k,d) (G). roblem.1 determine Π (k,d) (G) for a graph G..1 Smarandache path k-cover of a graph The Smarandache path k-cover of a graph was discussed by S. Arumugam and I.Sahul Hamid in [16]. Definition. A Smarandache path k-cover of a graph G is a Smarandache graphoidal (k, (G))-cover of G with =path for an integer k 1. A Smarandache path 1-cover of G such that its every edge is in exactly one path in it is called a simple path cover. The minimum cardinality of simple path covers of G is called the simple path covering number of G and is denoted by Π (1, (G)) (G). If do not consider the condition E(G i ) E(G j ) 1, then a simple path cover is called path cover of G, its minimum number of path cover is denoted by π(g) in reference. For examples, π s (K n ) = n and π s(t) = k, where k is the number of odd degree in tree T. roblem. determine Π (k,d) (G) for a graph G. Update Results for roblem. Obtained in [10]: (1) Π (1, (G)) (T) = π(t) = k, where k is the number of vertices of odd degree in T. () Let G be a unicyclic graph with cycle C. Let m denote the number of vertices of degree greater than on C. Let k be the number of vertices of odd degree. Then

14 Labeling, Covering and Decomposing of Graphs 11 Π (1, (G)) (G) = if m = 0 k + if m = 1 k + 1 if m = k if m () For a wheel W n = K 1 + C n 1, we have Π (1, (G)) (W n ) = 6 if n = 4 n + if n 5 roof Let V (W n ) = {v 0, v 1,..., v n 1 } and E(W n ) = {v 0 v i : 1 i n 1} {v i v i+1 : 1 i n } {v 1 v n 1 }. If n = 4, then W n = K 4 and hence Π (1, (G)) (W n )(W n ) = 6. Now, suppose n 5. Let r = n If n is odd, let i = (v i, v 0, v r+i ), i = 1,,..., r. r+1 = (v 1, v,..., v r ), r+ = (v 1, v r, v r 1,...,v r+ ) and r+ = (v r, v r+1, v r+ ). If n is even, let i = (v i, v 0, v r 1+i ), i = 1,,..., r 1. r = (v 0, v r 1 ), r+1 = (v 1, v,..., v r 1 ), r+ = (v 1, v r 1,...,v r+1 ) and r+ = (v r 1, v r, v r+1 ). Then Π (1, (G)) (W n ) = { 1,,..., r+ } is a simple path cover of W n. Hence π s (W n ) r + = n +. Further, for any simple path cover ψ of Wn at least three vertices on C = (v 1, v,..., v n 1 ) are terminal vertices of paths in ψ. Hence t q k, so that Π (1, (G)) (W n ) = q t k + = n (1, (G)) +. Thus Π (W n ) = n +. A. Nagarajan, V. Maheswari and S. Navaneethakrishnan discussed Smarandache path 1- cover in [17]. Definition. A Smarandache path 1-cover of G such that its every edge is in exactly two path in it is called a path double cover. Define G H with vertex set V (G) V (H) in which (g 1, h 1 ) is joined to (g, h ) whenever g 1 g E(G) or g 1 = g and h 1 h E(H); G H, the weak product of graphs G, H with vertex

15 1 Linfan Mao set V (G) V (H) in which two vertices (g 1, h 1 ) and (g, h ) are adjacent whenever g 1 g E(G) and h 1 h E(H) and γ (G) = min { ψ : ψ is a path double cover of G }. (4) Let m. if m is odd; γ (C m K ) = 6 if m is even. (5) Let m, n. γ (C m C n ) = 5 if at least one of the numbers m and n is odd. (6) Let m, n. 4 if n 1 or (mod 4) γ ( m C n ) = 8 if n 0 or (mod 4) (7) γ (C m K ) = 6 if m is odd. (8) γ ( m K ) = 4 for m. (9) γ ( m K ) = 5 for m. (10) γ (C m ) = 5 if m is odd. (11) γ ( m K ) = 4 for m. (1) γ (K m,n ) = max{m, n}. (1) if m= or n=; γ ( m n ) = 4 if m, n. (14) γ (C m C n ) = 5 if m, n and at least one of the numbers m and n is odd. (15) γ (C m K ) = 4 for m.. Smarandache graphoidal tree d-cover of a graph S.Somasundaram, A.Nagarajan and G.Mahadevan discussed Smarandache graphoidal tree d- cover of a graph in references [18]-[19]. Definition.4 A Smarandache graphoidal tree d-cover of a graph G is a Smarandache graphoidal ( G, d)-cover of G with =tree for an integer d 1. The minimum cardinality of Smarandache graphoidal tree d-cover of G is denoted by (G) = Π( G,d) (G). If d = (G), then γ (d) γ (d) T T (G) is abbreviated to γ T(G). roblem. determine γ T (G) for a graph G, particularly, γ T (G). Update Results for roblem. Obtained in [1-1]: Case 1: γ T (G) (1) γ T (K p ) = p ; () γ T (K m,n ) = m+n if m n < m. () γ T (K m,n ) = m if n > m.

16 Labeling, Covering and Decomposing of Graphs 1 (4) γ T ( m n ) = for integers m, n. (5) γ T ( n C m ) = for integers m, n. (6) γ T (C m C n ) = if m, n. Case : (1) γ (d) T (G) γ (d) T (K p) = p(p d+1) if d < p, p if d p if p 4. () γ (d) T (K m,n) = p + q pd = mn (m + n)(d 1) if n, m d. () γ (d) T (K d 1,d 1) = p + q pd = d 1. (4) γ (d) T (K n,n) = n for d n and n >. (5) γ (d) T (C m C n ) = for d 4 and γ () T (C m C n ) = q p. 5. Furthermore In fact, Smarandache s notion can be used to generalize more and more conceptions and problems in classical graph theory. Some of them will appeared in my books Automorphism Groups of Maps, Surfaces and Smarandache s Geometries (Second edition), Smarandache Multi- Space Theory (Second edition) published in forthcoming, or my monograph Graph Theory A Smarandachely Type will be appeared in 01. References [1] L.F.Mao, Combinatorial Geometry with Applications to Field Theory, InfoQuest, USA,009. [] L.F.Mao, Smarandache Multi-Space Theory, Hexis, hoenix, USA, 006. [] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research ress, 005. [4] L.F.Mao, Combinatorial Fields - An Introduction, International J.Math.Combin., Vol.,009, 1-. [5] ShreedharK, B. Sooryanarayana and Raghunath., Smarandachely k-constrained labeling of Graphs, International J.Math. Combin. Vol.1 (009), [6]. Devadas Rao, B. Sooryanarayana and M. Jayalakshmi, Smarandachely k-constrained Number of aths and Cycles, International J.Math. Combin. Vol. (009), [7] R. Vasuki and A. Nagarajan, Some Results on Super Mean Graphs, International J.Math. Combin. Vol. (009), [8] Bibin K. Jose, Open Distance-attern Uniform Graphs, International J.Math. Combin. Vol. (009), [9] Zhongfu Zhang et.al., On the AVSDT-Coloring of S m +W n, International J.Math. Combin. Vol. (008),

17 14 Linfan Mao [10] Jingwen Li, Zhongfu Zhang, Zhiwen Wang, Enqiang Zhu, Fei Wen and Bing Yao, The Smarandachely adjacent-vertex total coloring of three classes of -regular Halin graphs, Computational Intelligence and Software Engineering, 009. CiSE 009. International Conference on Issue Date: 11-1 Dec [11] Enqiang Zhu, Zhiwen Wang and Zhongfu Zhang, On the Smarandachely-adjacent-vertex edge coloring of some double graphs, J.Shandong University(Natural Science), 44(1),009. [1] Xunjing Lv, Smarandachely Adjacent Vertex Totol of Graph m + n, Ludong University Journal(Natural Science), 5(4), 009. [1] Shaowei Liang, On the Smarandachely Adjacent Vertex Total Coloring of k-cube Graph, Journal of Tangshan College, Vol., (009). [14] Shaowei Liang, On Smarandachely Adjacent Edge Coloring of a Class of Extended etersen Graph, J. Hebei North University(Natural Science), 5(5), 009. [15] Xiangen Chen and Zhongfu Zhang, Adjacent vertex distinguishing total coloring on m n, Journal of Northwest Normal University (Natural Science), Vol.41, 1(005). [16] S. Arumugam and I. Sahul Hamid, Simple path covers in graphs, International J.Math. Combin. Vol. (008), [17] A. Nagarajan, V. Maheswari and S. Navaneethakrishnan, ath double covering number of product graphs, International J.Math. Combin. Vol. (009), 7-8. [18] S.Somasundaram, A.Nagarajan and G.Mahadevan, Decomposition of graphs into internally disjoint trees, International J.Math. Combin. Vol. (009), [19] S.Somasundaram, A.Nagarajan and G.Mahadevan, Graphoidal Tree d - Cover, International J.Math. Combin. Vol. (009),