On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs

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Abstract

Let G be a tripartite graph with tripartition $$(V_{1},V_{2},V_{3})$$, where $$\mid V_{1}\mid =\mid V_{2}\mid =\mid V_{3}\mid =k>0$$. It is proved that if $$d(x)+d(y)\ge 3k$$ for every pair of nonadjacent vertices $$x\in V_{i}, y\in V_{j}$$ with $$i\ne j(i,j\in \{1,2,3\})$$, then G contains k vertex-disjoint triangles. As a corollary, if $$d(x)\ge \frac{3}{2}k$$ for each vertex $$x\in V(G)$$, then G contains k vertex-disjoint triangles. Based on the above results, vertex-disjoint triangles in multigraphs are studied. Let M be a standard tripartite multigraph with tripartition $$(V_{1},V_{2},V_{3})$$, where $$\mid V_{1} \mid =\mid V_{2}\mid =\mid V_{3} \mid =k>0$$. If $$\delta (M)\ge 3k-1$$ for even k and $$\delta (M)\ge 3k$$ for odd k, then M contains k vertex-disjoint 4-triangles $$\varDelta _{4}$$(a triangle with at least four edges). Furthermore, examples are given showing that the degree conditions of all our three results are best possible.

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Acknowledgements

We are grateful to the referees for the valuable suggestions and hints that helped to improve this article.

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Correspondence to Qingsong Zou.

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This work was supported by National Natural Science Foundation of Shaanxi Province (Grant no. 2020JM-199) and National Natural Science Foundation of China (Grant no. 11401455)

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Zou, Q., Li, J. & Ji, Z. On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs. Graphs and Combinatorics 36, 1355–1361 (2020). https://doi.org/10.1007/s00373-020-02188-3

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• DOI: https://doi.org/10.1007/s00373-020-02188-3