# Embedding of a Signed Graph with Property \(P\) in a Graceful Signed Graph with Property \(P\)

### Synopsis

Let \(S = (V, E, s)\) be a signed graph with \(|V|=p, |E|=q\) and let \(s: E \rightarrow\{+, -\}\) be a function which assigns a sign *+* or *-* to each edge. For any injection \(f: V \rightarrow\{0, 1, …, q\}\), the induced edge labelling *g _{f}* is defined by \(g_f(uv)=s(uv)|f(u)-f(v)|\). The function

*f*is said to be a graceful labelling of

*S*, if \(g_f(E^+) = \{1, 2, …, |E^+|\}\) and \(g_f(E^-) = \{-1, -2, …, -|E^-|\}\) where E

^{+}and E

^{-}denote the set of all positive and negative edges of

*S*respectively. A signed graph that admits graceful labelling is called a graceful signed graph. In this paper, we prove that a signed graph

*S*having property

*P*can be embedded in a graceful signed graph

*S'*having property

*P*when

*P*denotes the property being: triangle-free, planar, Eulerian, or Hamiltonian. We have also proved that if

*S*is a connected graph and

*S*is its decomposition into edge-induced subgraphs with \(f: V \rightarrow N \cup \{0\} \) an injection having maximum vertex label

_{1}, S_{2}, … S_{k}*M*, such that the edge-induced function g

_{S(f)}_{f}assigns distinct labels to edges of \(S_i, 1 \leq i \leq k\). Then

*S*can be embedded as an induced subgraph of

*k*-hypergraceful eulerian graph

*S'*with

*k*‑hypergraceful labelling

*h*such that \(M_{S'(h)} \leq 2^{k+1}(M_{S(f)}+4)-7\).

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