Complement Crown Reduction in Graph Theory

• Complement crown reduction is a structural principle in graph theory that transfers chromatic bounds in graphs to clique covering bounds in their complements.
• It establishes tight inequalities, such as deriving g(x) = ⌊8/5·x⌋ from a constant chromatic function, to rigorously relate graph and complement parameters.
• These insights guide algorithm design in graph coloring and covering and open avenues for exploring the limits of complementary-bounded functions.

Complement crown reduction is a structural principle in graph theory that describes how properties connected to coloring and clique-covering bounds transfer from a graph to its complement. Specifically, it addresses the relationship between $\chi$-boundedness (chromatic number bound) in a given graph class and $\theta$-boundedness (clique covering number bound) in its complement class, often expressed as tight inequalities or optimal bounding functions. This concept plays a central role in understanding nearly perfect graphs and in the paper of minimal reductions that preserve both coloring and clique structure through graph complementation.

1. Definitions and Key Concepts

A class $\mathcal{C}$ of graphs is $\chi$-bounded if there exists a function $f$ such that every graph $G \in \mathcal{C}$ satisfies $\chi(G) \leq f(\omega(G))$, where $\chi(G)$ is the chromatic number and $\omega(G)$ is the clique number. An “$f\chi$-bounded” class is one admitting such a function $f$ as a global bound across all induced subgraphs. The function $f$ encapsulates the tightness between local structure (maximum clique size) and global coloring requirements.

For a graph $G$, the complement $\overline{G}$ is formed by converting edges to nonedges and vice versa. There is a well-documented duality: coloring in a graph translates to clique covering in its complement, with $\theta(G)$ (the minimum number of cliques covering $V(G)$) playing an analogous role to $\chi(G)$. If a class is $\chi$-bounded by $f$, the natural inquiry is whether the complement class is similarly bounded, often by a related function $g$, termed a “complementary-bounded function.”

2. Principal Results

The work “Complements of nearly perfect graphs” (Gyárfás et al., 2013) presents several strong results concerning complement crown reduction:

• If $\mathcal{C}$ is $\chi$-bounded by the constant function $f(x) = 3$, the class of complements $\overline{\mathcal{C}}$ is $\chi$-bounded by $g(x) = \left\lfloor \frac{8}{5}x \right\rfloor$, and this bound is best possible.
• For each constant $c > 0$, if $f(x) = x$ for all $x \geq c$, then the class of complements is $\chi$-bounded.
• There exist families of graphs $\chi$-bounded by $f(x) = x + \frac{x}{\log^j(x)}$ whose complements are not $\chi$-bounded for any function, demonstrating the limits of complementary-boundedness for even modest deviations from the identity bounding function.
• Extending the classic Perfect Graph Theorem, every eventually identity function (i.e., $f(x) = x$ for all $x \geq c$) is complementary-bounded; this asserts solidity for broad classes of “almost perfect” graphs under complement crown reduction.

3. Structural and Algorithmic Implications

The complement crown reduction principle yields deep structural insight into the symmetry of graph classes under complementation. The precise characterization of when chromatic number and clique covering bounds remain tight across complements directly influences algorithm design for coloring and covering problems. The case $f(x) = 3$, $g(x) = \left\lfloor \frac{8}{5}x \right\rfloor$ also connects to classical theorems, such as Kőnig’s theorem and the Perfect Graph Theorem, and employs advanced combinatorial methods including color-critical graphs and Gallai’s results on factor-critical graphs.

A plausible implication is that any algorithm exploiting complement crown reduction to bound chromatic or clique covering numbers must abide by the explicit function transfer characterized by these results. Reductions relying on “crown-like” structures or subsampling would be forced within the optimal bounds provided ($\left\lfloor \frac{8}{5}x \right\rfloor$ for $f(x) = 3$), and not exceed them.

4. Examples and Counterexamples

A fundamental example is the graph $G_{5,8}$ with $|V(G_{5,8})| = 15$, $\chi(G_{5,8}) = 3$, $\alpha(G_{5,8}) = 5$, and $\theta(G_{5,8}) = 8$. This showcases the tightness of the bound: $$\theta(G) \leq \left\lfloor \frac{8}{5} \alpha(G) \right\rfloor$$.

By constructing disjoint copies and small modifications of $G_{5,8}$, the optimality of this bound for arbitrary values of $\alpha(G)$ is established, as every integer $x$ can be realized via a graph $G$ with $\alpha(G) = x$ and $$\theta(G) = \left\lfloor \frac{8}{5}x \right\rfloor$$.

As a counterexample, families of graphs built from Schrijver graphs (subgraphs of Kneser graphs) are crafted to realize $\chi$-boundedness by $f(x) = x + \frac{x}{\log^j(x)}$, whose complements defy $\chi$-boundedness by any function. This marks a sharp boundary: only very tightly “near-perfect” classes respect the complement crown reduction transfer.

5. Connections to Crown Reduction Procedures

The principle underlying complement crown reduction also appears in orthogonal representation constructions for minimum semidefinite rank, such as in the complement of cactus graphs (Navarro, 2018). Here, the orthogonal representation in $\mathbb{R}^5$ ensures that the minimum semidefinite rank of the complement is tightly bounded by a uniform dimension, and the inductive addition of vertices with strictly controlled nonadjacency mimics crown reduction procedures.

The strategy is to systematically grow the complement graph such that the “crown” (the substructure with exceptional adjacency patterns) is minimized at each step, hence controlling the semidefinite rank and regaining tight bounds after the reduction. This approach is relevant for establishing uniform results for broader graph families.

6. Open Problems and Future Directions

The complement crown reduction framework leaves several open avenues:

• Whether $f(x) = x + 1$ is complementary-bounded remains unresolved, representing the minimal unsolved case for “almost perfect” bounding functions.
• Investigations into more robust graph classes or hypergraph coverings using crown reduction techniques promise gains in bounding chromatic and clique-covering constraints across complements.
• Extending these principles, especially via construction methods that avoid increasing the nonadjacency or rank, could impact both theoretical and algorithmic domains in extremal and structural graph theory.

7. Summary and Significance

Complement crown reduction delineates the transfer of tight coloring and covering bounds between a $\chi$-bounded class and its complement, specializing in nearly perfect and robustly structured graphs. The discovery that $f(x) = 3$ yields the optimal complementary bound $g(x) = \left\lfloor \frac{8}{5}x \right\rfloor$, that eventually identity functions preserve this property, and that minor deviations can break it provides a comprehensive framework for understanding chromatic-clique covering duality and reduction methodology in graph complementation. Both abstract and constructive techniques—such as those using Schrijver graphs and crown-based inductive constructions—solidify these results and guide future inquiry at the intersection of coloring, covering, and graph structure.