On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs

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.