Representation Number of a Caterpillar

For a finite graph \(G\) with vertices \(\{v_1, \ldots, v_r\}\), a representation of \(G\) modulo \(n\) is a set \(\{a_1, \ldots, a_r\}\) of distinct, nonnegative integers with \(0 \leq a_i < n\), satisfying \(\gcd(a_i – a_j, n) = 1\) if and only if \(v_i\) is adjacent to \(v_j\). The representation number, \(Rep(G)\), is the smallest \(n\) such that \(G\) has a representation modulo \(n\). Evans \(et \, al.\) obtained the representation number of paths. They also obtained the representation number of a cycle except for cycles of length \(2^k + 1\), \(k \geq 3\). In the present paper, we obtain upper and lower bounds for the representation number of a caterpillar, and get its exact value in some cases.