The Edge-graceful Spectra of Connected Bicyclic Graphs Without Pendant

Abstract

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, Lee, Cheng, and Wang 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.