Improved Upper Bounds on $a'(G\Box H)$

The acyclic edge colouring problem is extensively studied in graph theory. The corner-stone of this field is a conjecture of Alon et. al.\cite{alonacyclic} that $a'(G)\le \Delta(G)+2$. In that and subsequent work, $a'(G)$ is typically bounded in terms of $\Delta(G)$. Motivated by this we introduce a term $gap(G)$ defined as $gap(G)=a'(G)-\Delta(G)$. Alon's conjecture can be rephrased as $gap(G)\le2$ for all graphs $G$. In \cite{manusccartprod} it was shown that $a'(G\Box H)\le a'(G)+a'(H)$, under some assumptions. Based on Alon's conjecture, we conjecture that $a'(G\Box H)\le a'(G)+\Delta(H)$ under the same assumptions, resulting in a strengthening. The results of \cite{alonacyclic} validate our conjecture for the class of graphs it considers. We prove our conjecture for a significant subclass of sub-cubic graphs and state some generic conditions under which our conjecture can be proved. We suggest how our technique can be potentially applied by future researchers to expand the class of graphs for which our conjecture holds. Our results improve the understanding of the relationship between Cartesian Product and acyclic chromatic index.