Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two.
A tree containing exactly two non-pendant vertices is called a double-star. A double-star with degree sequence (k1 + 1, k2 + 1, 1, . . . , 1) is denoted by Sk1,k2. We study the edge-decomposition of graphs into double-stars. It was proved that every double-star of size k decomposes every 2k-regular graph. In this paper, we extend this result by showing that every graph in which every vertex has degree 2k + 1 or 2k + 2 and containing a 2-factor is decomposed into Sk1,k2 and Sk1−1,k2, for all positive...
Let be a graph, and the smallest integer for which has a nowhere-zero -flow, i.e., an integer for which admits a nowhere-zero -flow, but it does not admit a -flow. We denote the minimum flow number of by . In this paper we show that if and are two arbitrary graphs and has no isolated vertex, then except two cases: (i) One of the graphs and is and the other is -regular. (ii) and is a graph with at least one isolated vertex or a component whose every block is an...
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