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Crystal graphs are powerful combinatorial tools for working with the placticmonoid and symmetric functions. Quasi-crystal graphs are an analogous conceptfor the hypoplactic monoid and quasi-symmetric functions. This paper makes acombinatorial study of these objects. A previously-observed isomorphism ofcomponents of the quasi-crystal graph is explained and explicitly describedusing a new combinatorial structure called a quasi-array. Then two conjecturesof Maas-Gari\'epy on the interaction of fundamental quasi-symmetric functionsand Schur functions and on the arrangement of quasi-crystal components withincrystal components are settled, the former positively, the latter negatively.

Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions