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The edge-graceful spectra of connected bicyclic graphs without pendant. (English) Zbl 1160.05041

Summary: Let \(G\) be a connected simple \((p,q)\)-graph and \(k\) a non-negative integer. The graph \(G\) is said to be \(k\)-edge-graceful if the edges can be labeled with \(k,k+1,\dots,k+q-1\) so that the vertex sums are distinct modulo \(p\). The set of all \(k\) where \(G\) is \(k\)-edge-graceful is called the edge-graceful spectrum of \(G\). In 2004, S.-M. Lee, K.-J. Cheng and Y.-C. Wang [”On the edge-graceful spectra of cycles with one chord and dumbbell graphs,” Congr. Numeratium 170, 171–183 (2004; Zbl 1066.05131)] analyzed the edge-graceful spectra of certain connected bicyclic graphs, leaving some cases as open problems. Here, we determine the edge-graceful spectra of all connected bicyclic graphs without pendant.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C38 Paths and cycles

Citations:

Zbl 1066.05131
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