Chromatic Number
(graph/chromatic_number.hpp)

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• Last update: 2026-08-13 01:41:40+09:00
• Include: #include "graph/chromatic_number.hpp"

Overview

Computes the chromatic number of a graph: the minimum number of colors needed to color all vertices so that adjacent vertices have different colors.

The algorithm is intended for small general graphs. It uses inclusion-exclusion over vertex subsets and supports at most 20 vertices.

Graph Interpretation

Every active edge of Graph<T> is treated as an undirected edge, regardless of how it was inserted. Parallel edges do not change the answer. Self-loops are ignored, matching the conventions of the other exact vertex-set algorithms in this library.

Function

Function	Signature	Description	Complexity
chromatic_number	template <class T> int chromatic_number(const Graph<T>& g)	Returns the chromatic number. The empty graph has chromatic number 0.	$O(N2^N)$ time and $O(2^N)$ memory

The function asserts that g.size() <= 20.

Algorithm

For every vertex subset S, the implementation first counts the independent subsets contained in S. Inclusion-exclusion then counts ordered covers of all vertices by k independent sets. Such a cover exists exactly when the graph is k-colorable.

The count is evaluated modulo 14 pairwise coprime numbers. For N, k <= 20, the number of covers is less than 2^(N k) <= 2^400, while the product of the moduli is greater than 2^412. Thus a positive count cannot vanish modulo all of them, and the result is deterministic.

Example

#include "graph/chromatic_number.hpp"
#include "graph/graph.hpp"
#include <iostream>

int main() {
    m1une::graph::Graph<> g(5);
    for (int v = 0; v < 5; v++) g.add_edge(v, (v + 1) % 5);

    std::cout << m1une::graph::chromatic_number(g) << "\n";  // 3
}

Depends on

• Graph (graph/graph.hpp)

Required by

• Graph All (graph/all.hpp)
• Undirected Graph Algorithms (graph/undirected.hpp)

Verified with

• verify/graph/chromatic_number.test.cpp
• verify/graph/chromatic_number_randomized.test.cpp
• verify/graph/cow_game.test.cpp
• verify/graph/graph_algorithms.test.cpp
• verify/graph/range_edge_graph.test.cpp

Code

#ifndef M1UNE_GRAPH_CHROMATIC_NUMBER_HPP
#define M1UNE_GRAPH_CHROMATIC_NUMBER_HPP 1

#include <array>
#include <bit>
#include <cassert>
#include <cstdint>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

namespace detail {

struct ChromaticResidues {
    static constexpr std::array<std::uint32_t, 14> mod = {
        1000000007, 1000000009, 998244353, 985661441, 943718401, 935329793, 918552577,
        897581057,  880803841,  754974721, 645922817, 595591169, 469762049, 167772161,
    };

    std::array<std::uint32_t, 14> value;

    explicit ChromaticResidues(std::uint32_t x = 0) {
        value.fill(x);
    }

    void multiply(std::uint32_t x) {
        for (int i = 0; i < int(mod.size()); i++) {
            value[i] = std::uint32_t(std::uint64_t(value[i]) * x % mod[i]);
        }
    }
};

}  // namespace detail

template <class T>
int chromatic_number(const Graph<T>& g) {
    int n = g.size();
    assert(n <= 20);
    if (n == 0) return 0;

    std::vector<std::uint32_t> adjacent(n, 0);
    for (const auto& e : g.edges()) {
        if (e.from == e.to) continue;
        adjacent[e.from] |= std::uint32_t(1) << e.to;
        adjacent[e.to] |= std::uint32_t(1) << e.from;
    }

    std::uint32_t subset_count = std::uint32_t(1) << n;
    std::vector<std::uint32_t> independent_count(subset_count, 0);
    independent_count[0] = 1;
    for (std::uint32_t mask = 1; mask < subset_count; mask++) {
        int v = std::countr_zero(mask);
        std::uint32_t rest = mask ^ (std::uint32_t(1) << v);
        independent_count[mask] =
            independent_count[rest] + independent_count[rest & ~adjacent[v]];
    }

    std::vector<detail::ChromaticResidues> power(subset_count, detail::ChromaticResidues(1));
    for (int colors = 1; colors <= n; colors++) {
        std::array<std::uint32_t, 14> sum = {};
        for (std::uint32_t mask = 0; mask < subset_count; mask++) {
            power[mask].multiply(independent_count[mask]);
            bool positive = ((n - std::popcount(mask)) & 1) == 0;
            for (int i = 0; i < int(sum.size()); i++) {
                std::uint32_t x = power[mask].value[i];
                if (positive) {
                    sum[i] += x;
                    if (sum[i] >= detail::ChromaticResidues::mod[i]) {
                        sum[i] -= detail::ChromaticResidues::mod[i];
                    }
                } else {
                    sum[i] = (sum[i] >= x ? sum[i] - x
                                          : sum[i] + detail::ChromaticResidues::mod[i] - x);
                }
            }
        }
        for (std::uint32_t x : sum) {
            if (x != 0) return colors;
        }
    }
    return n;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_CHROMATIC_NUMBER_HPP

#line 1 "graph/chromatic_number.hpp"



#include <array>
#include <bit>
#include <cassert>
#include <cstdint>
#include <vector>

#line 1 "graph/graph.hpp"



#line 6 "graph/graph.hpp"
#include <utility>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 11 "graph/chromatic_number.hpp"

namespace m1une {
namespace graph {

namespace detail {

struct ChromaticResidues {
    static constexpr std::array<std::uint32_t, 14> mod = {
        1000000007, 1000000009, 998244353, 985661441, 943718401, 935329793, 918552577,
        897581057,  880803841,  754974721, 645922817, 595591169, 469762049, 167772161,
    };

    std::array<std::uint32_t, 14> value;

    explicit ChromaticResidues(std::uint32_t x = 0) {
        value.fill(x);
    }

    void multiply(std::uint32_t x) {
        for (int i = 0; i < int(mod.size()); i++) {
            value[i] = std::uint32_t(std::uint64_t(value[i]) * x % mod[i]);
        }
    }
};

}  // namespace detail

template <class T>
int chromatic_number(const Graph<T>& g) {
    int n = g.size();
    assert(n <= 20);
    if (n == 0) return 0;

    std::vector<std::uint32_t> adjacent(n, 0);
    for (const auto& e : g.edges()) {
        if (e.from == e.to) continue;
        adjacent[e.from] |= std::uint32_t(1) << e.to;
        adjacent[e.to] |= std::uint32_t(1) << e.from;
    }

    std::uint32_t subset_count = std::uint32_t(1) << n;
    std::vector<std::uint32_t> independent_count(subset_count, 0);
    independent_count[0] = 1;
    for (std::uint32_t mask = 1; mask < subset_count; mask++) {
        int v = std::countr_zero(mask);
        std::uint32_t rest = mask ^ (std::uint32_t(1) << v);
        independent_count[mask] =
            independent_count[rest] + independent_count[rest & ~adjacent[v]];
    }

    std::vector<detail::ChromaticResidues> power(subset_count, detail::ChromaticResidues(1));
    for (int colors = 1; colors <= n; colors++) {
        std::array<std::uint32_t, 14> sum = {};
        for (std::uint32_t mask = 0; mask < subset_count; mask++) {
            power[mask].multiply(independent_count[mask]);
            bool positive = ((n - std::popcount(mask)) & 1) == 0;
            for (int i = 0; i < int(sum.size()); i++) {
                std::uint32_t x = power[mask].value[i];
                if (positive) {
                    sum[i] += x;
                    if (sum[i] >= detail::ChromaticResidues::mod[i]) {
                        sum[i] -= detail::ChromaticResidues::mod[i];
                    }
                } else {
                    sum[i] = (sum[i] >= x ? sum[i] - x
                                          : sum[i] + detail::ChromaticResidues::mod[i] - x);
                }
            }
        }
        for (std::uint32_t x : sum) {
            if (x != 0) return colors;
        }
    }
    return n;
}

}  // namespace graph
}  // namespace m1une

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