Some Irregular Total Labelings of Expansion Graphs expan(P_m, C_n)

摘要

For a simple graph G = (V(G),E(G)) with vertex set V(G) and edge set E(G), a total labeling λ : V(G)∪E(G)→{1,2,...,k} is called an edge-irregular total k-labeling of G if for any two different edges e = e_1e_2 and f = f_1 f_2 in E(G), we have wt(e) + wt(f), where wt(e) = λ(e_1) + λ(e) + λ(e_2). Meanwhile, a total labeling θ : V(G)∪E(G) → {1,2,...,k} is called a vertex-irregular total k-labeling of G if for any two different vertices u and v in V(G), we obtain wt(u) + wt(v), where wt(u) = {formula}. The minimum value of k for which there exists an edge (a vertex)-irregular total k-labeling of G is called the total edge (vertex) irregular strength of G, denoted by tes(G) (tvs(G)). In this paper, we consider an expansion graph expan(P_m, C_n), where P_m is a path on m vertices and C_n is a cycle on n vertices. An expan(P_m, C_n) is a graph obtained from a copy of P_m and m + n copies of C_n by sticking the i-th copy of C_n at i-th vertex of P_m and sticking the j-th copy of C_n at the j-th edge of P_m. We determine tes(expan(P_m, C_n)) and tvs(expan(P_m, C_n)) for any integers m ≥ 2 and n ≥ 3.