Computes the exact chromatic number of a graph via inclusion-exclusion for graphs up to 20 vertices, helping determine the minimum colors needed without overlap.
Compute the exact chromatic number of a graph using inclusion-exclusion (up to 20 vertices).
| Name | Required | Description | Default |
|---|---|---|---|
| edges | Yes | Edges as [u, v] pairs (0-indexed) | |
| vertex_count | Yes | Number of vertices (max 20) |
Changes observed during successful MCP inspections.
v0.1.11Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?
No annotations are provided, so the description bears the full burden. It usefully discloses the algorithm (inclusion-exclusion, which is exponential) and the 20-vertex cap, giving the agent a sense of cost and limits. It omits failure behavior above the cap and whether the result is a single integer.
Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.
Is the description appropriately sized, front-loaded, and free of redundancy?
A single front-loaded sentence that conveys purpose, algorithm, and constraint with zero filler. Nothing is redundant and the key limitation appears immediately.
Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.
Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?
For a parameterizable math tool with a rich schema and no output schema, the description covers the core intent and the vertex cap. But it omits runtime/complexity implications, what happens when the cap is exceeded, and how this differs operationally from sibling coloring tools, leaving gaps an agent may need.
Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.
Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?
Schema coverage is 100%, so both parameters (vertex_count, edges) are already fully documented in the schema. The description adds the 'max 20 vertices' bound, which restates the schema's own 'max 20' note rather than extending it. Baseline 3 is appropriate.
Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.
Does the description clearly state what the tool does and how it differs from similar tools?
States a precise verb+resource ('Compute the exact chromatic number of a graph') and even names the underlying algorithm (inclusion-exclusion). It is clearly distinguishable from the many graph siblings like graph_coloring (which is heuristic) and bipartite_check (which only tests 2-colorability).
Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.
Does the description explain when to use this tool, when not to, or what alternatives exist?
The 'exact' + 'up to 20 vertices' constraint implies when this tool is appropriate (small graphs needing a guaranteed answer) and by extension when to prefer a faster heuristic. However, it does not explicitly name graph_coloring as the alternative for large graphs or state the failure mode when vertex_count exceeds 20.
Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.