關於(2,3)-圖形零和流數之研究

Abstract

對一無向圖形 G，令 E(v) 記為圖形中頂點 v 的相鄰邊所構成之集合。圖 G 上一零和流為一組對邊的非零實數編號 f 使得對每一頂點 v 來說，

∑ f (e) = 0 e∈E(v)

皆成立。 零和 k-流為一零和流且編號全來自集合{±1,...,±(k−1)}。 零和流數 F(G) 定義為圖 G 具有零和 k-流之最小正整數 k。在此篇論文中，對一(2,3)-圖形 G 給出了具有零和流數 3 的充分且必要之條件。此外我們研究由路徑和樹擴展而成之(2,3)-圖形上的零和流數，名曰，聖誕燈、樹燈，並總結它們的零和流數最多為 5。

For an undirected graph G, let E(v) denote the set of edges incident on a vertex v ∈ V(G). A zero-sum flow is an assignment f of non-zero real numbers on the edges of G such that

∑ f (e) = 0 e∈E(v)

for all v ∈ V(G). A zero-sum k-flow is a zero-sum flow with integers from the set {±1,...,±(k−1)}. Let zero-sum flow number F(G) be defined as the least number of k such that G admits a zero-sum k-flow. In this paper, a necessary and sufficient condition for (2,3)-graph G with F(G) = 3 is given. Furthermore we study zero-sum flow number of (2,3)-graphs expanded from path and tree, namely, the Christmas lamps, the tree lamps, respectively, and conclude that their zero-sum flow numbers are at most 5.