Prime Labelings of Snake Graphs

A prime labeling of a graph G with n vertices is a labeling of the vertices with distinct integers from the set {1, 2 ,..., n} such that the labels of any two adjacent vertices are relatively prime. In this paper, we introduce a snake graph, the fused union of identical cycles, and define a consecutive snake prime labeling for this new family of graphs. We characterize some snake graphs that have a consecutive snake prime labeling and then consider a variation of this labeling

Prime labelings on planar grid graphs

It is known that for any prime p and any integer n such that 1≤n≤p there exists a prime labeling on the pxn planar grid graph PpxPn. We show that PpxPn has a prime labeling for any odd prime p and any integer n such that that p\u3cn≤p2

On Prime Labelings of Uniform Cycle Snake Graphs

A prime labeling of a graph of order n is an assignment of the integers 1, 2, ... , n to the vertices such that each pair of adjacent vertices has coprime labels. For positive integers m, k, q with k  ≥  3 and 1  ≤  q   ≤ ⌊k/2⌋ , the uniform cycle snake graph Cmk,q is constructed by taking a path with m edges and replacing each edge by a k-cycle by identifying two vertices at distance q in the cycle with the vertices of the original path edge. We construct prime labelings for Cmk,q for many pairs (k,q) and, in each case, all m. These include: all cases with k  ≤   9 or k = 11; all cases with q = 2 when k ≡ 3 (mod 4); all cases with q = 3 when k has the form 2a − 1, 3a + 1 or 3b + 3 for all a and all odd b; all cases with q = 4 when k is even; and all cases with q = k/2 when q is a prime congruent to 1 (mod 3)