A vertex of degree one is called an end-vertex and the set of end-vertices of G is denoted by End(G). For a positive integer k, a tree T be called k-ended tree if |End(T )| ≤ k. In this paper, we obtain sufficient conditions for spanning k-trees of 3-regular connected graphs. We give a construction sequence of graphs satisfying the condition. At the end, we present a conjecture about spanning k-ended trees of 3-regular connected graphs.
We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.2010 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 11C20 15A15.
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.2000 Mathematics Subject Classification. 05A18, 11B73. Key words and phrases. Multiplicative partition function; Stirling numbers of the second kind; mixed partitions of a set.
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