Compatible spanning circuits in edge-colored graphs

NEW

Problem: Reconstructed formal question: Does every finite simple edge-colored 2-connected graph $G$ on $n$ vertices satisfying Fan’s condition

contain a compatible spanning circuit, i.e. a closed trail visiting every vertex and with consecutive edges of distinct colors, such that each vertex $v$ is visited at least

times?

This is the natural universal reading of the problem statement. The wording “under what conditions” is open-ended, but under the stated hypotheses alone the assertion is false.

Result: Counterexample.

Let $G=K_{3,3}$ with bipartition

Then $n=6$, every vertex has degree $3$, and every pair of vertices at distance $2$ lies in the same part, so

Thus $G$ is 2-connected and satisfies Fan’s condition.

Color the edges red/blue as follows:

all other edges are blue.

Each vertex is incident with both colors, so this is not merely a monochromatic obstruction.

Since $d(v)=3$ for every vertex,

Thus any compatible spanning circuit would have to visit every vertex at least once.

Suppose such a circuit $C$ existed. A circuit is a closed trail, so the used edges form a connected even spanning subgraph $H\subseteq K_{3,3}$. Since every vertex of $K_{3,3}$ has degree $3$, every vertex of $H$ must have even positive degree, hence degree exactly $2$. Therefore $H$ is a Hamiltonian $6$-cycle.

Because only two colors are used, compatibility forces the colors around this $6$-cycle to alternate. Hence each vertex is incident in $H$ with exactly one red edge and one blue edge. Therefore the blue edges of $H$ form a perfect matching of $K_{3,3}$.

But the blue subgraph has edge set

It has no perfect matching: the two vertices $b_1,b_2$ have only one blue neighbor, namely $a_3$. This violates Hall’s condition.

Therefore no compatible spanning circuit exists. Hence the stated Fan-type hypotheses alone do not guarantee the desired compatible spanning circuit.

Audit: The graph is finite, simple, 2-connected, satisfies Fan’s condition exactly, and the required visit lower bound is $1$ at every vertex. The conclusion fails completely: there is no compatible spanning circuit at all.

Citation: No external citation used; the disproof is the explicit counterexample above.