(k+1)-kernels and the number ofk-kings in

Rui Xia Wang

¹

Abstract

a b s t r a c t Let D = (V (D), A(D)) be a digraph and k ≥ 2 be an integer. A vertex x is a k-king of D, if for every y. . x k of length k, x 0 and x k are adjacent. Recently, we have shown that a k-quasi-transitive digraph with k ≥ 4 has a kking if and only if it has a unique initial strong component D 1 and D 1 is not isomorphic to an extended (k + 1)-cycle. In this article, we will study the number of k-kings in a k-quasitransitive digraph when it has a k-king. Indeed, we show that when k = 4, it may have exactly one 4-king; if k ≥ 5, then it has at least two k-kings. In addition, we obtain new results on the minimum number of (k + 1)-kings in k-quasi-transitive digraphs.Galeana-Sánchez et al. conjectured that every k-quasi-transitive digraph has a (k + 1)-kernel. In this article, we shall prove that the conjecture is true.

k‐quasi‐transitive digraphs of large diameter

“…We will begin by showing that every vertex outside a path of length k + 2 realizing the distance between its extremes, must be adjacent to some vertices in the path, and there are four possibilities for these adjacencies. The former statement is an immediate consequence of the following result found in [10].…”

“…Lemma 6 (Wang and Meng [10]). Let k be an integer such that ≥ k 5, let D be a strong kquasi-transitive digraph and let C be a cycle of length n with ≥ n k − 1.…”

“…So, the existing results actually show that, for each integer greater than 2, the class of ‐quasi‐transitive digraphs generalizes some property of quasi‐transitive digraphs. Galeana‐Sánchez and Hernández‐Cruz introduced ‐quasi‐transitive digraphs in [9] while studying ‐kernels, and later, the same authors along with Juárez‐Camacho [7], and also Wang with different coauthors [10, 12] considered the existence and number of ‐kings in ‐quasi‐transitive digraphs. Although many structural results were developed to obtain good generalizations of the results known for 3‐kings in quasi‐transitive digraphs, no tool as powerful as the Canonical Decomposition is yet known for ‐quasi‐transitive digraphs.…”

Hamilton cycle problem in strong k-quasi-transitive digraphs with large diameter

“…A digraph is k-quasi-transitive, if for any path x 0 x 1 · · · x k of length k, x 0 and x k are adjacent. The k-quasi-transitive digraphs have been studied in [2][3][4][5][6][7].…”

-panchromatic digraphs

“…Several sufficient conditions for the existence of k-kernels in nearly tournaments have been proved. For instance, see [4], [28], [30] and [39].…”