For d-degenerate C4-free graphs, the fractional chromatic number is at most (1+o(1))d/log d, and for every fixed girth at least 4 there are d-degenerate graphs with fractional chromatic number at least (1−o(1))d/log d.
Coloring graphs with forbidden bipartite sub- graphs
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Sharp bounds for the fractional chromatic number of high-girth $d$-degenerate graphs
For d-degenerate C4-free graphs, the fractional chromatic number is at most (1+o(1))d/log d, and for every fixed girth at least 4 there are d-degenerate graphs with fractional chromatic number at least (1−o(1))d/log d.