Degeneracy bounds, stability, and a sharp gap for $B$-colorings

A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $Δ_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\le d\leΔ$, every finite simple $d$-degenerate graph $G$ with $Δ(G)\leΔ$ satisfies $$q_B(G)\le Δ+(d-1)Δ_2(G)\le dΔ.$$ Consequently, $dΔ$ is the exact maximum, with equality precisely for graphs containing $K_{d,Δ}$. More generally, if $q_B(G)\ge dΔ-s$, where $0\le s<Δ$, then $G$ contains $K_{d,Δ-s}$; if also $s<d$, then $G$ has at least $d-s$ vertices of degree $Δ$ with the same open neighborhood. For $Δ\ge3$, we further show that every $K_{3,Δ}$-free 3-degenerate graph satisfies $q_B(G)\le3Δ-2$; the example $K_{3,Δ-1}$ shows that this bound is best possible up to one. For loopless multigraphs, we establish a sharp gap in the possible values of $q_B(G)$. For every integer $Δ\ge3$, every finite loopless multigraph $G$ with $Δ(G)\leΔ$ satisfies $$q_B(G)\leΔ(Δ-1)$$ unless $G$ has a component isomorphic to $K_{Δ,Δ}$, in which case $q_B(G)=Δ^2$. The bound $Δ(Δ-1)$ is attained by both $K_{Δ,Δ-1}$ and $K_{Δ,Δ}-e$. Consequently, among finite loopless multigraphs with maximum degree at most $Δ$, no value of $q_B(G)$ lies strictly between $Δ^2-Δ$ and $Δ^2$.