Graph Coloring Calculator

Graph coloring fundamentals for five-vertex graphs

Graph coloring on this page is a way to organize five vertices that have pairwise conflicts. A graph has vertices and edges; whenever two vertices are joined by an edge, they cannot receive the same color label. A proper vertex coloring assigns labels so neighboring vertices differ, and the fewest labels that can satisfy every edge is the graph's chromatic number. This calculator solves that precise problem from a 5×5 adjacency matrix.

For a five-vertex graph, this calculator can search for an exact answer rather than estimate one. When it reports a chromatic number of 3, the search has failed with 1 and 2 colors and succeeded with 3. The displayed assignment therefore connects the formal graph-theory definition to a usable coloring of the vertices you entered.

Formally, the graph is written as , and the quantity being found is the chromatic number . Within the five-vertex limit, you can model empty and complete graphs, paths, stars, odd cycles, triangles, and other small dense graphs.

How to use the 5-vertex graph coloring calculator

To color a five-vertex graph, enter its adjacency matrix carefully before starting the search. Each matrix cell represents a pair of vertices: 1 means an edge is present and 0 means no edge is present. Click Color Graph after filling the matrix. The result gives the chromatic number followed by one color label for each vertex.

1. Choose the graph to test, with vertices numbered 0 through 4.
2. Represent its edges with a 5×5 matrix of 0s and 1s.
3. For an undirected graph, use 0 on the diagonal and match each entry with its reflection across the diagonal.
4. Click Color Graph to obtain the minimum color count and one valid coloring.

The output values 1, 2, 3, and so on are category labels rather than literal paint colors. Depending on the application, they can stand for exam periods, radio channels, seating groups, or registers. The graph-coloring requirement remains unchanged: endpoints of an edge must receive different labels.

Entering a five-vertex adjacency matrix

For this graph coloring calculator, the adjacency matrix is , with an entry set to 1 when vertices i and j are connected and 0 otherwise. For a simple undirected graph, three input checks make the result straightforward to interpret.

• Diagonal entries should be 0, because a simple graph has no self-loops: a00 = a11 = a22 = a33 = a44 = 0.
• Symmetry should hold, so aij = aji. If you enter a13 = 1, then a31 should also be 1.
• Only 0 and 1 are intended. Empty fields are interpreted as 0 by the script, but clean 0/1 input is best.

For an asymmetric matrix, do not interpret the result as the usual undirected graph chromatic number. Vertices are assigned in order 0 through 4, so for a pair of distinct vertices i < j, the conflict check encountered when vertex j is assigned uses aji. A symmetric matrix ensures that every undirected edge is represented consistently.

Chromatic-number definition and exact backtracking method

A proper coloring of the entered graph is a function that gives every vertex one of labels. It is valid only when each edge has differently colored endpoints: . The chromatic number is the least for which such a coloring exists.

The calculator applies that definition by searching in increasing color-count order. It attempts one color first, then two, then three, continuing through five if necessary. For a proposed value of , it assigns vertices from 0 to 4. A candidate label is rejected when it duplicates the label of an already assigned adjacent vertex. If a later assignment becomes impossible, the search removes the earlier choice and tries another label. This try, recurse, and undo process is backtracking.

Five-vertex backtracking algorithm overview

The implementation uses isSafe(v, color, adj, colors) to test whether a proposed label for vertex v duplicates the color of an adjacent vertex that has already been placed. The recursive routine assign(v, maxColors, adj, colors) tests labels 1 through maxColors, recurses after every safe choice, and clears a choice if the deeper call cannot finish the coloring. When all five vertices have assignments, the current color vector is a valid result.

Checking candidate counts from smallest to largest is what makes the displayed value a chromatic number rather than merely the size of an arbitrary coloring. The calculator finds a valid partition and establishes that every smaller number of color labels failed for the matrix supplied.

Worked example: a five-vertex graph containing triangles

This example graph contains triangles, so it needs at least three colors. Enter the matrix below and click Color Graph; the calculator reports a chromatic number of 3.

One valid output coloring for that graph is . Thus vertex 0 has color 1, vertex 1 has color 2, vertex 2 has color 3, vertex 3 has color 1, and vertex 4 has color 2. Another solver may show different label numbers, which is still correct if adjacent vertices differ and the number of labels remains minimal.

Interpreting a graph coloring result

The graph coloring result has two useful parts: the chromatic number states the minimum number of compatible groups, and the per-vertex table gives one way to form those groups. When checking a hand-drawn graph, compare the table with the edges in the matrix to confirm that every connected pair has different labels.

Five-vertex graph families to test

Testing familiar five-vertex graphs makes the chromatic number easier to recognize from structure. The fixed size is useful for experimentation because a single changed edge can be entered quickly and its effect on the exact answer can be observed immediately.

• No edges (empty graph): every off-diagonal entry is 0, so all vertices can share one color and χ(G) = 1.
• Complete graph K5: every off-diagonal entry is 1, so each vertex conflicts with all the others and χ(G) = 5.
• Path graph P5: connect 0–1–2–3–4 and keep symmetry. Paths are bipartite, so χ(G) = 2.
• Cycle graph C5: connect 0–1–2–3–4–0. Because the cycle length is odd, two colors fail and χ(G) = 3.
• Complete bipartite K2,3: split the vertices into {0,1} and {2,3,4}, then connect every cross pair. Bipartite graphs need only two colors, so χ(G) = 2.

Five-vertex input assumptions and limits

• Fixed size: this interface is built specifically for 5 vertices. The script reads a00 through a44.
• Input values: use 0 or 1 for clear adjacency-matrix interpretation. Empty cells are treated as 0.
• Undirected graphs: standard vertex coloring uses a symmetric adjacency matrix.
• Color labels: the output labels are abstract categories, not literal hues.
• Optimality: because the search attempts smaller values first, the first successful color count is the chromatic number for a valid simple undirected input.

Why five-vertex graph coloring matters

Graph coloring matters because it turns exclusion constraints into a compact mathematical model. Exams sharing students cannot occupy the same time slot; nearby transmitters should not share a frequency; simultaneously live compiler variables cannot share a register. In each case, vertices represent items, edges represent conflicts, and colors represent the limited resources assigned without conflict.

This calculator deliberately uses a small graph, but it demonstrates the challenge behind larger coloring problems. At five vertices, the edges and exact answer are easy to inspect. That foundation helps explain why larger graphs often use greedy methods, DSATUR-style heuristics, integer programming, or specialized solvers rather than uncomplicated exhaustive search.

Reliable adjacency-matrix entry tips

For dependable graph coloring results, inspect the matrix before questioning the search. A missing reflected entry means the matrix is not an undirected graph description, and a diagonal 1 is outside the simple-graph setup used by standard vertex coloring. Sketching the graph from the matrix and checking each edge against the table is an effective way to catch entry mistakes.

The calculator returns only one minimum coloring. A textbook answer can use another numbering of the same color classes and still be mathematically identical. Swapping the labels 1 and 2 changes neither the edge constraints nor the chromatic number; the important tests are different labels on adjacent vertices and the minimum total number of labels.