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We report some novel properties of a square lattice filled with white sites,randomly occupied by black sites (with probability $p$). We considerconnections up to second nearest neighbours, according to the following rule.Edge-sharing sites, i.e. nearest neighbours of similar type are alwaysconsidered to belong to the same cluster. A pair of black corner-sharing sites,i.e. second nearest neighbours may form a 'cross-connection' with a pair ofwhite corner-sharing sites. In this case assigning connected status to bothpairs simultaneously, makes the system quasi-three dimensional, withintertwined black and white clusters. The two-dimensional character of thesystem is preserved by considering the black diagonal pair to be connected witha probability $q$, in which case the crossing white pair of sites are deemeddisjoint. If the black pair is disjoint, the white pair is consideredconnected. In this scenario we investigate (i) the variation of the Eulernumber $\chi(p) \ [=N_B(p)-N_W(p)]$ versus $p$ graph for varying $q$, (ii)variation of the site percolation threshold with $q$ and (iii) sizedistribution of the black clusters for varying $p$, when $q=0.5$. Here $N_B$ isthe number of black clusters and $N_W$ is the number of white clusters, at acertain probability $p$. We also discuss the earlier proposed 'Island-Mainland'transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90:213) and show mathematically that the proposed transition is not, in fact, acritical phase transition and does not survive finite size scaling. It is alsoexplained mathematically why clusters of size 1 are always the most numerous.

Euler Number and Percolation Threshold on a Square Lattice with Diagonal Connection Probability and Revisiting the Island-Mainland Transition