# Achromatic Coloring on Double Star Graph Families

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A Smarandachely k-constrained labeling of a graph G(V,E) is a bijective mapping f : V ? E ? {1, 2, .., |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 ? 2. A graph G which admits a such...

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Let G be a graph and f : V (G) ? {1, 2, 3, ï¿½, p + q} be an injection. For each edge e = uv and an integer m ? 2, the induced Smarandachely edge m-labeling f*S is defined by f*S(e) = ?f(u) + f(v)/m? Then f is called a Smarandachely super m-mean labeling if f(V (G)) ? {f*(e) : e ? E(G)} = {1,...

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Let G = (V,E) be a graph. A subset S of V is called a Smarandachely degree equitable k-set for any integer k, 0 ? k ? ?(G) if the degrees of any two vertices in S differ by at most k. It is obvious that S = V (G) if k = ?(G). A Smarandachely degree equitable 1-set is usually called a degree...

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Let G1 = (V1,E1), G2 = (V2,E2) be two graphs. For a chosen edge set E ? E2, the Smarandache VE-product G1 ï¿½VE G2 of G1, G2 is defined by V (G1 ï¿½VE G2) = V1 ï¿½ V2, E(G1 ï¿½VE G2) = {(a, b)(a', b')|a = a', (b, b') ? E2, or b = b', (a, a') ? E1} ? {(a, b)(a', b')|(a, a') ? E1 and...

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Given an arbitrary non-empty subset M of vertices in a graph G = (V,E), each vertex u in G is associated with the set fï¿½M(u) = {d(u, v) : v ? M, u ? v}, called its open M-distance-pattern. A graph G is called a Smarandachely uniform k-graph if there exist subsets M1,M2, ï¿½ï¿½ï¿½...

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A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G, s) (S = (G, ï¿½)) where G = (V, E) is a graph called underlying graph of S and s : E ? (e1, e2, ï¿½, ek) (ï¿½ : V ? (e1, e2, ï¿½, ek)) is a function, where each ei ? {+, -}. Particularly, a...

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A Smarandachely k-signed digraph (Smarandachely k-marked digraph) is an ordered pair S = (D, s) (S = (D, ï¿½)) where D = (V, A) is a digraph called underlying digraph of S and ï¿½ : A ? (e1, e2; ï¿½, ek) (ï¿½ : V ? (e1, e2, ï¿½, ek)) is a function, where each ei ? {+, -}....

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No abstract available.

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A Smarandache drawing of a graph G is a drawing of G on the plane with minimal intersections for its each component and a circulant graph C(n; S) is the graph with vertex set V (C(n; S)) = {viâˆ£0 â©½ i â©½ n-1} and edge set E(C(n; S)) = {vivj âˆ£0 â‰¤ i â‰ j â‰¤ n-1,...