On 2-Y-homogeneous $$(Y,Y')$$-distance-biregular graphs with $$D=4$$

Distance-biregular graphs form a natural bipartite generalization of distance-regular graphs. Among them, the class of 2- Y -homogeneous distance-biregular graphs plays a prominent role and has been the subject of several recent classification efforts. In this paper, we complete the classification of 2- Y -homogeneous $$(Y,Y')$$ ( Y , Y ′ ) -distance-biregular graphs with eccentricity $$D=4$$ D = 4 . Building on earlier work that settled the cases c₂'=1 c 2 ′ = 1 and c₂'=2 c 2 ′ = 2 , we address the remaining open case c₂'≥ 3 c 2 ′ ≥ 3 . We prove that no such graphs exist, thereby resolving an open problem posed in previous work and closing the classification program for eccentricity $$D=4$$ D = 4 . Our approach combines a detailed analysis of intersection numbers, arithmetic constraints arising from 2- Y -homogeneity, and structural properties of distance-biregular graphs. As a consequence, we obtain a complete characterization of all 2- Y -homogeneous distance-biregular graphs with $$D=4$$ D = 4 , and we further conclude that every 2- Y -homogeneous distance-biregular graph with c₂'≥ 3 c 2 ′ ≥ 3 must necessarily have eccentricity $$D=3$$ D = 3 .