The spiders $S(4m+2,\,2m,\,1)$ are e-positivite

We establish the e-positivity of spider graphs of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, Wang and van Willigenburg. A key to our proof is the eI-expansion formula of the chromatic symmetric function of paths due to Shareshian and Wachs, where the symbol~I indicates integer compositions rather than partitions. Following the strategy of the divide-and-conquer technique, we pick out one or two positive eJ-terms for each negative eI-term in an e-expression for the spiders, where J are selected to be distinct compositions obtained by rearranging the parts of I.