This paper investigates the enumeration of rooted nearly 2-regular planar maps,provides the enumerating equations satisfied by enumerating functions with the valency of root-face,and the number of edges as two parameters.Some explicit expressions of them are also derived.
Abstract Let and denote the minimum size of a decycling set and maximum genus of a graph G , respectively. For a connected cubic graph G of order n , it is shown that . Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k ‐regular graph G is , where c and are the number of components of and the number of edges in , respectively. Therefore, S is minimum if and only if is minimum. As an application, this leads to a lower bound for of a k ‐regular graph G . In many cases this bound may be sharp.
Abstract: Although much work has been done on enumerating rooted planar maps since Tutte’s pioneeering works in early 1960s, many classes of maps with no loops are still untreated. In this paper, we enumerate rooted loopless eulerian planar maps with three parameters: the valency of rooted vertex, the number of nonrooted vertices and the number of inner faces.
This article first uses the matroid language to difine the new concepts of graph as the new concepts of matroid, and then we use the matroid language to translate the new results which Goddyn and Heuvel have attained in graph into the new results of matroid. Last, we will prove these new results by the method of matroid.
This work provided enumerating equations satisfied by enumerating functions with valency of root-face,size,number of non-root-vertices as parameters.Moreover,some explicit expressions of these functions were also derived.
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