# Graceful labelings of the generalized Petersen graphs

Document Type : Original paper

Authors

1 University of Maribor

2 School of Information Science &amp; 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

Abstract

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 ${u_1, u_2, cdots, u_n} cup {v_1, v_2, cdots, v_n}$ 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.

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