Is this a bug between RelationGraph and BipartiteGraphQ?

Question

Inspired by yode's latest question I was playing around with RelationGraph and found that BipartiteQ seems to be unable to detect certain bipartite results.

As an example, I was looking at the graph where each vertex is a 1- or 2-digit integer, and edges indicate whether a 2-digit integer starts or ends with a 1-digit integer. This is clearly a bipartite graph (and GraphLayout -> "BipartiteEmbedding" shows this), since there are no edges between 2-digit numbers and no edges between single digits. And up to n = 19 this works fine:

g = RelationGraph[(StringEndsQ[#1, #2] || StringStartsQ[#2, #])
   && # != #2 &, ToString /@ Range[0, 19], 
 GraphLayout -> "BipartiteEmbedding", VertexLabels -> "Name"]

BipartiteGraphQ @ g
(* True *)

But as soon as I get to 20:

g = RelationGraph[(StringEndsQ[#1, #2] || StringStartsQ[#2, #])
   && # != #2 &, ToString /@ Range[0, 20], 
 GraphLayout -> "BipartiteEmbedding", VertexLabels -> "Name"]

BipartiteGraphQ @ g
(* False *)

The problem seems to be something internal to the result of RelationGraph, since extracting the edges and reconstructing the graph works:

BipartiteGraphQ @ Graph @ EdgeList @ g
(* True *)

We can however localise the problem a bit by computing the graph on a smaller set of vertices. It's possible to remove all but 6 vertices/edges and still reproduce the problem:

g = RelationGraph[(StringEndsQ[#1, #2] || StringStartsQ[#2, #])
   && # != #2 &, ToString /@ {0, 1, 2, 10, 12, 20}, 
 GraphLayout -> "BipartiteEmbedding", VertexLabels -> "Name"]

BipartiteGraphQ @ g
(* False *)

Removing any further edges "resolves" the problem.

Is this a bug in BipartiteGraphQ? Am I misunderstanding something about bipartite graphs?

I'm using Mathematica 10.4 on Windows 10.