Sequential Graph

A Connected Graph having $e$ Edges is said to be sequential if it is possible to label the nodes $i$ with distinct Integers $f_i$ in $\{0, 1, 2, \ldots, e-1\}$ such that when Edge $ij$ is labeled $f_i+f_j$, the set of Edge labels is a block of $e$ consecutive integers (Grace 1983, Gallian 1990). No Harmonious Graph is known which cannot also be labeled sequentially.

See also Connected Graph, Harmonious Graph

References

Gallian, J. A. ``Open Problems in Grid Labeling.'' Amer. Math. Monthly 97, 133-135, 1990.

Grace, T. ``On Sequential Labelings of Graphs.'' J. Graph Th. 7, 195-201, 1983.