The Graceful Coalescence of Alpha Cycles

Year 2019, Volume: 2 Issue: 2, 114 - 120, 27.06.2019

Christian Barrientos Sarah Minion

https://doi.org/10.33434/cams.505485

Abstract

The standard coalescence of two graphs is extended, allowing to identify two isomorphic subgraphs instead of a single vertex. It is proven here that any succesive coalescence of cycles of size $n$, where $n$ is divisible by four, results in an $\alpha$-graph, that is, the most restrictive kind of graceful graph, when the subgraphs identified are paths of sizes not exceeding $\frac{n}{2}$. Using the coalescence and another similar technique, it is proven that some subdivisions of the ladder $L_n = P_2 \times P_n$ also admit an $\alpha$-labeling, extending and generalizing the existing results for this type of subdivided graphs.

Keywords

Coalescence, $\alpha$-labeling, Graceful labeling, Ladder

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