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.