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We explore the physical properties of the completely packed O($n$) loop modelon the square lattice, and its generalization to an Eulerian graph model, whichfollows by including cubic vertices which connect the four incoming loopsegments. This model includes crossing bonds as well. Our study of theproperties of this model involve transfer-matrix calculations and finite-sizescaling. The numerical results are compared to existing exact solutions,including solutions of special cases of a so-called coloring model, which areshown to be equivalent with our generalized loop model. The latter exactsolutions correspond with seven one-dimensional branches in the parameter spaceof our generalized loop model. One of these branches, describing the case ofnonintersecting loops, is already known to correspond with the orderingtransition of the Potts model. We find that another exactly solved branch,which describes a model with nonintersecting loops and cubic vertices,corresponds with a first-order Ising-like phase transition for $n>2$. For$1<n<2$, this branch can be interpreted in terms of a low-temperature O($n$)phase with corner-cubic anisotropy. For $n>2$ this branch is the locus of afirst-order phase boundary between a phase with a hard-square lattice-gas likeordering, and a phase dominated by cubic vertices. The first-order character ofthis transition is in agreement with a mean-field argument.

Completely packed O($n$) loop models and their relation with exactly solved coloring models