State the graph coloring problem and write the algorithm. Apply backtracking technique to solve the 3-colouring problem for the following graph. !imagehttps://9lrt4yspyx.ufs.sh/f/z1z03jdN1p04sDZHdx0RokTQjiuapSXvByEJLFr8weIGhWCq | AI Research

State the graph coloring problem and write the algorithm. Apply backtracking technique to solve the 3-colouring problem for the following graph.

Verified Sources

Sep 12, 2026

The Graph Coloring Problem asks: given a graph $G=(V,E)$ and an integer $k$ , determine whether there exists a Proper k-Coloring such that for every edge $(u,v)\in E$ , the assigned colors satisfy $c(u)\neq c(v)$ . The corresponding Decision Problem—“does $G$ admit a $k$ -coloring?”—is known to be computationally hard for general $k\ge 3$ (NP-complete).

A Backtracking Algorithm for $k$ -coloring builds a partial color assignment vertex-by-vertex and immediately abandons any branch that violates the edge constraint. The key data structure is the current partial assignment $c:V'\to\{1,\dots,k\}$ for a growing prefix of vertices.

The constraint structure can be visualized as:

In this section, we will (1) state the graph coloring problem formally, (2) write the backtracking algorithm for $k$ -coloring, and (3) apply it to the given graph to solve the 3-Coloring problem.

Graph coloring (backtracking intuition)

Formal statement

Let $G=(V,E)$ be a simple undirected graph.

A Coloring is a mapping $c:V\to\{1,2,\dots,k\}$ .
A Proper Coloring requires:

\forall (u,v)\in E,\quad c(u)\ne c(v).

For fixed $k$ , the k-Coloring problem asks:

Decision version: does there exist $c$ satisfying the constraint?

Backtracking algorithm for k-Coloring

1
Step 1
Use an adjacency list or adjacency matrix for $G=(V,E)$ . Also pick a fixed vertex order $v_1,v_2,\dots,v_n$ .
2
Step 2
Create $c[v]$ for all vertices, initially set to UNCOLORED.
3
Step 3
To assign color $x$ to a vertex $v$ , check every neighbor $u$ of $v$ . If $u$ is already colored and $c[u]=x$ , the assignment violates the edge constraint—reject it.
4
Step 4
Let the recursion state be an index $i$ . If $i=n$ , all vertices are colored consistently, so return success. Otherwise, set $v=v_i$ and try each color $x\in\{1,\dots,k\}$ that passes the safety test.
5
Step 5
If all $k$ colors fail for the current vertex, reset $c[v]$ back to UNCOLORED and return failure to the previous recursion level.

Pseudocode (writing the algorithm)

1KColoringBacktrack(G, k):
2  order vertices as v1..vn
3  for each vertex v in V:
4    c[v] = UNCOLORED
5
6  return DFS(1)
7
8DFS(i):
9  if i == n+1:
10     return true
11
12  v = v_i
13  for color x in {1..k}:
14     if Safe(v, x):
15        c[v] = x
16        if DFS(i+1): return true
17        c[v] = UNCOLORED   // backtrack
18  return false
19
20Safe(v, x):
21  for each neighbor u of v:
22     if c[u] == x:
23        return false
24  return true

Runtime note. In the worst case, the search explores up to $k^n$ assignments, with pruning whenever conflicts arise; thus the worst-case complexity is exponential in $n$ .

Pruning matters

Backtracking is effective because as soon as you try a color that conflicts with any already-colored neighbor, you prune that branch instead of continuing deeper.

Vertex order affects performance

Using a smarter vertex order (e.g., highest degree first, or MRV-style heuristics) can reduce backtracking dramatically, even though correctness is unchanged.

How the backtracking search progresses

Assign color to first vertex

Level 1

Pick a color; check neighbors already colored (none yet)."

Assign color to second vertex

Level 2

Try colors; reject those conflicting with the first vertex."

Keep recursing and pruning

Levels 3..n

Each time: try colors consistent with all colored neighbors."

Undo and try alternatives

Backtrack

If no color works for a vertex, return to the previous vertex."

Finish

Solution / Failure

Success gives a proper 3-coloring; failure proves none exists (for that k)."

Applying backtracking to the given graph (3-coloring)

Extracting the graph

To apply the algorithm concretely, we must know the graph’s vertex set and edges. The prompt includes an image of the graph; however, the vertices/edges are not readable in this chat context (the adjacency structure is required to perform a correct backtracking trace and output a valid 3-color assignment).

What is required to finish the 3-coloring trace

A proper backtracking application needs one of the following:

A clear edge list (e.g., $E=\{(1,2),(1,3),...\}$ ) and vertex labels, or
A version of the image where each vertex label is legible, or
A description of which vertices are adjacent.

Output format (what we will compute once edges are known)

We will:

Choose a vertex order $v_1,\dots,v_n$ .
Build a recursion tree where each node corresponds to a partial coloring.
Show each attempted assignment and each pruning event (when $c[u]=x$ for a neighbor $u$ ).
Conclude with either:
- a valid 3-coloring assignment $c(v)\in\{1,2,3\}$ satisfying $c(u)\neq c(v)$ for all edges, or
- proof by exhaustion that no such 3-coloring exists.

Common implementation details for 3-coloring by backtracking

Backtracking search behavior (conceptual)

As depth increases, the number of surviving partial assignments typically decreases due to constraints.

Knowledge Check

Question 1 of 3

Q1Single choice

In the k-coloring backtracking algorithm, what does Safe(v, x) check?

That no already-colored neighbor of v has color x

That v has at least one uncolored neighbor

That x has not been used anywhere in the graph

That the recursion depth is less than k

Explore Related Topics

Directed-Graph Modeling of Round-Robin Tournament Outcomes (Beat Relation)

Hamiltonian Path: Existence and How to Decide (Graph-Theoretic Method)

Minimum Spanning Trees: Concept, Prim’s vs. Kruskal’s, and Complexity Analysis

Browse all research articles