| Summary: | 碩士 === 中原大學 === 應用數學研究所 === 100 === Let G=(V,E) be a graph. A (2,1)-total labeling of G is a mapping from VUE into {0,…, λ} for some integer λ such that (i) if x and y are adjacent vertices, then l(x) =/=l(y);(ii) if e and ｆ are adjacent edges, then l(e)=/=l(f);(iii) if an edge e is incident to a vertex x, then |l(e)-l(x)|>=2. The span of a (2,1)-total labeling is the maximum difference between two labels. The (2,1)-total number of a graph G is the minimum span of a (2,1)-total labeling of G, denoted by λ(G).
In this thesis, we mainly focus on the (2,1)-total labelings of 3-chromatic cubic graphs, and we prove the following results:
1.Let n be an integer and n>=2. For i=1,2,...,n, let B_i=(U_i,V_i) be the complete bipartite K_3,3 and let e_i1 and e_i2 be non-adjacent edges in B_i. Then λ(e_11(B_1)e_12-e_21(B_2)e_22-...-e_n1(B_n)e_n2-e_11(B_1)e_12)=5.
2.Let G=(A,B) be a (3,l)-crown ,where A={a_1,a_2,...,a_l} and B={b_1,b_2,...,b_l}. Then λ(π(G;k))=5, fork=2,3,...[l/2]+1.
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