An Improved Bound for Equitable Proper Labellings

For every graph G with size m and no connected component isomorphic to \(K_2\) , we prove that, for \(L=(1,1,2,2,\dots ,\lfloor m/2 \rfloor +2,\lfloor m/2 \rfloor +2)\) , we can assign labels of L to the edges of G in an injective way so that no two adjacent vertices of G are incident to the same sum of labels. This implies that every such graph with size m can be labelled in an equitable and proper way with labels from \(\{1,\dots ,\lfloor m/2 \rfloor +2\}\) , which improves on a result proved by Haslegrave, and Szabo Lyngsie and Zhong, implying this can be achieved with labels from \(\{1,\dots ,m\}\) .