Graceful labelings of the generalized Petersen graphs

Document Type : Original paper


1 University of Maribor

2 School of Information Science & Technology, Chengdu University, Chengdu, China

3 College of Information Science and Technology, Chengdu University of Technology, Chengdu, China

4 Key Laboratory of High Confidence Software Technologies, Peking University, Peking, China


A graceful labeling of a graph $G=(V,E)$ with $m$ edges is  an injection $f: V(G) \rightarrow \{0,1,\ldots,m\}$ such that the resulting edge labels obtained by $|f(u)-f(v)|$ on every edge $uv$ are pairwise distinct.  For natural numbers $n$ and  $k$, where $n > 2k$, a generalized Petersen graph $P(n, k)$ is the graph whose vertex set is $\{u_1, u_2, \ldots,  u_n\}  \cup \{v_1, v_2, \ldots, v_n\}$ and its edge set is  $\{u_iu_{i+1}, u_iv_i, v_iv_{i+k} :  1 \leq i \leq n \}$, where subscript arithmetic is done modulo $n$. We propose a backtracking algorithm with a specific static variable ordering and dynamic  value ordering to find graceful labelings for generalized Petersen graphs. Experimental results show that the  presented approach strongly outperforms the standard backtracking algorithm. The proposed algorithm is able to find  graceful labelings for all generalized Petersen graphs $P(n, k)$ with $n \le 75$ within only several seconds. 


Main Subjects

[1] G.S. Bloom and S.W. Golomb, Applications of numbered undirected graphs, Proceedings of the IEEE 65 (1977), no. 4, 562–570.
[2] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2009), no. 6, 1–219.
[3] T.A. Redl, Graceful graphs and graceful labelings: two mathematical programming formulations and some other new results, Congr. Numer. 164 (2003), 17–32.
[4] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, 1966, pp. 349–355.
[5] A. Steimle and W. Staton, The isomorphism classes of the generalized petersen graphs, Discrete Math. 309 (2009), no. 1, 231–237.