A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em&g...A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em>k</em></sub>} of directed cycles in <i>H</i> such that these directed cycles together cover all vertices in <i>H</i> and such that the arc sets of these directed cycles induce a connected subdigraph of <i>H</i>. A subdigraph <i>F</i> of a digraph <i>D</i> is a circulation if for every vertex in <i>F</i>, the indegree of <em>v</em> equals its out degree, and a spanning circulation if <i>F</i> is a cycle factor. Define <i>f</i> (<i>D</i>) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from <i>D</i> by contracting all arcs in <i>F</i>, among all circulations <i>F</i> of <i>D</i>. Adigraph <i>D</i> is supereulerian if <i>D</i> has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> are nontrivial strong digraphs such that <em>D</em><sub>1</sub> is supereulerian and <em>D</em><sub>2</sub> has a cycle vertex cover C’ with |C’| ≤ |<em>V</em> (<em>D</em><sub>1</sub>)|, then the Cartesian product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is also supereulerian. In this paper, we prove that for strong digraphs<em> D</em><sub>1</sub> and <em>D</em><sub>2</sub>, if for some cycle factor <em>F</em><sub>1</sub> of <em>D</em><sub>1</sub>, the digraph formed from <em>D</em><sub>1</sub> by contracting arcs in F1 is hamiltonian with <i>f</i> (<i>D</i><sub>2</sub>) not bigger than |<em>V</em> (<em>D</em><sub>1</sub>)|, then the strong product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is supereulerian.展开更多
A digraph D is oriented if it does not contain 2-cycles. If an oriented digraph D has a directed eulerian path, it is an oriented eulerian digraph. In this paper, when an oriented eulerian digraph D has minimum out-de...A digraph D is oriented if it does not contain 2-cycles. If an oriented digraph D has a directed eulerian path, it is an oriented eulerian digraph. In this paper, when an oriented eulerian digraph D has minimum out-degree 2 and a diameter d, we find the minimum order of D. In addition, when D is 2-regular with diameter 4rn （m≥2）, we classify the extremal cases.展开更多
文摘A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em>k</em></sub>} of directed cycles in <i>H</i> such that these directed cycles together cover all vertices in <i>H</i> and such that the arc sets of these directed cycles induce a connected subdigraph of <i>H</i>. A subdigraph <i>F</i> of a digraph <i>D</i> is a circulation if for every vertex in <i>F</i>, the indegree of <em>v</em> equals its out degree, and a spanning circulation if <i>F</i> is a cycle factor. Define <i>f</i> (<i>D</i>) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from <i>D</i> by contracting all arcs in <i>F</i>, among all circulations <i>F</i> of <i>D</i>. Adigraph <i>D</i> is supereulerian if <i>D</i> has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> are nontrivial strong digraphs such that <em>D</em><sub>1</sub> is supereulerian and <em>D</em><sub>2</sub> has a cycle vertex cover C’ with |C’| ≤ |<em>V</em> (<em>D</em><sub>1</sub>)|, then the Cartesian product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is also supereulerian. In this paper, we prove that for strong digraphs<em> D</em><sub>1</sub> and <em>D</em><sub>2</sub>, if for some cycle factor <em>F</em><sub>1</sub> of <em>D</em><sub>1</sub>, the digraph formed from <em>D</em><sub>1</sub> by contracting arcs in F1 is hamiltonian with <i>f</i> (<i>D</i><sub>2</sub>) not bigger than |<em>V</em> (<em>D</em><sub>1</sub>)|, then the strong product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is supereulerian.
基金Supported by the University of Incheon Research Grant in 2009-2010
文摘A digraph D is oriented if it does not contain 2-cycles. If an oriented digraph D has a directed eulerian path, it is an oriented eulerian digraph. In this paper, when an oriented eulerian digraph D has minimum out-degree 2 and a diameter d, we find the minimum order of D. In addition, when D is 2-regular with diameter 4rn （m≥2）, we classify the extremal cases.
