L(p, q)-labelings of subdivision of graphs

• Main Authors: Meng-Hsuan Tsai, 蔡孟璇
• Other Authors: David Kuo
• Format: Others
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/19084079405949355416

碩士 === 國立東華大學 === 應用數學系 === 101 === Given a graph G and a function h from E(G) to N, the h-subdivision of G, denoted by G_{(h)}, is the graph obtained from G by replacing each edge uv in G with a path P:ux_{uv}¹x_{uv}²…x_{uv}ⁿ⁻¹v, where n=h(uv). When h(e)=c is a constant for all e∈E(G), we use G_{(c)} to replace G_{(h)}. Given a graph G and two positive integers p,q with p>q, an L(p,q)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|≥p if d_{G}(x,y)=1, and |f(x)-f(y)|≥q if d_{G}(x,y)=2. A k-L(p,q)-labeling is an L(p,q)-labeling such that no label is greater than k. The L(p,q)-labeling number of G, denoted by λ_{p,q}(G), is the smallest number k such that G has a k-L(p,q)-labeling. We study the L(p,q)-labeling numbers of subdivisions of graphs in this thesis. We give an upper bound of λ_{p,q}(G₍₃₎) for those graphs G with Δ(G)≥3, and show that λ(G₍₃₎)=Δ+1 for any graph G with Δ(G)≥4. We also show that λ(G_{(h)})=Δ+1 if Δ(G)≥5 and h is a function from E(G) to N so that h(e)≥3 for all e∈E(G), or if Δ(G)≥4 and h is a function from E(G) to N so that h(e)≥4 for all e∈E(G).