Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 A Note on Distance-Fall Colorings 1157 1161 15088 10.22049/cco.2026.30962.2683 EN Wayne Goddard School of Mathematical and Statistical Sciences Clemson University, USA Sonwabile Mafunda Soka University of America, USA University of Johannesburg, South Africa Journal Article 2025 08 29 We say a proper coloring  of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, then it has a distance-2 fall 3-coloring. Further, for every integer $k\ge 2$, if $T$ is a tree of order at least $k$, then $T$ has a $k$-coloring such that every vertex is within distance $k-1$ of every color. This proves an old conjecture of Beineke and Henning that every tree of order $n$ has an independent distance-$d$-dominating set of size at most $n/(d + 1)$. https://comb-opt.azaruniv.ac.ir/article_15088_335ba1e7ff9537a3161ff540c7c78ce7.pdf