On Harmonious Labeling of Graphs

• Main Authors: Shang-Yuan Wang, 王尚元
• Other Authors: Chin-Lin Shiue
• Format: Others
• Language: en_US
• Published: 2005
• Online Access: http://ndltd.ncl.edu.tw/handle/32496893885778728398

碩士 === 中原大學 === 應用數學研究所 === 93 === A function f is called a harmonious labeling of a graph G with q edges if it is an injection from the vertex set of G into { 0,1,2,…,q-1} such that when each edge {x,y} is assigned the label f(x)+f(y)(mod q), the resulting edge labels are distinct. In case that G is a tree, exactly one label may be used on two vertices. A k-star of length m is a graph obtained from a star with k edges by replacing each edge with a path of length m, denoted by S(k,m). I. Cahit [2] proved that a k-star S(k,m) has a harmonious labeling except for even values of m when k is not equal to 2 . Now we consider a graph obtained from a S(k,m) by identifying all the pendent vertices. Such a graph is called hammock, denoted by Hk,m. In this thesis we first show that a hammock graph Hk,m has no harmonious labelings when m is odd and k=2(mod 4), and then we prove that Hk,m has a harmonious labeling when both k and m are odd.