Linear arboricity of the tensor products of complete multipartite graphs
DOI:
https://doi.org/10.55016/ojs/cdm.v20i2.77218Abstract
The linear arboricity of a graph $G,$ denoted by $\ell a(G),$ is the minimum number of linear forests which partition the edge set of $G.$ Akiyama et al. conjectured that $\ell a(G)=\left\lceil\frac{k+1}{2} \right\rceil$ for any $k$-regular graph $G.$ This conjecture is proved to be true for $k=3, 4, 5, 6, 8, 10.$ Also, in [P. Paulraja and S. Sivasankar, Linear Arboricity of the Tensor Products of Graphs, Utilitas Math. 99 (2016) 295--317], the conjecture is proved for the tensor product of complete graphs. Although the conjecture was not proved in general, we have proved that the tensor product of two regular complete multipartite graphs confirms the conjecture in the affirmative.
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