In the book Research Problems in Discrete Geometry, on page 252, it is stated that every K_4 free graph is a subgraph of the odd distance graph. We just proved that W_5, the five wheel is not a subgraph of the odd distance graph. I further believe that there are triangle free graphs that are not subgraphs of the odd distance graph, and even graphs with large girth.
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CSI of Charles University
The Odd-Distance Graph
In the book Research Problems in Discrete Geometry, on page 252, it is stated that every K_4 free graph is a subgraph of the odd distance graph. We just proved that W_5, the five wheel is not a subgraph of the odd distance graph. I further believe that there are triangle free graphs that are not subgraphs of the odd distance graph, and even graphs with large girth.