On a problem of Sivaraman and a problem of Gyárfás

Abstract: The \textit{girth} of a graph $G$, denoted $\mathrm{g}(G)$, is the length of a shortest cycle in $G$. If $G$ contains no cycle, we define $\mathrm{g}(G)=\infty$. Sivaraman (2020) asked for the optimal $χ$-bounding function for the class of graphs whose complements have girth at least $6$. Let (F(s) = \max{χ(G): ω(G)\le s,\ \mathrm{g}(\overline{G})\ge 6}). We prove that there exists a constant (c>0) such that [ c\left(\frac{s}{\log s}\right)^{4/3} \le F(s) \le (1+o(1))\frac{s^{3/2}}{\log s}. ] For small values, we establish the exact results [ F(1)=1,\; F(2)=2,\; F(3)=4,\; F(4)=5,\; F(5)=6,\; F(6)=8, ] and each bound is sharp. A graph $G$ is \emph{almost perfect} if every induced subgraph $H$ of $G$ satisfies (α(H)ω(H)+1\ge |V(H)|). Gyárfás (2023) asked whether almost perfect graphs are $χ$-bounded by the function $g(x)=x+1$. We answer this question in the negative by showing that there is no constant $c$ such that every almost perfect graph $G$ satisfies $χ(G)\le ω(G)+c$.

• The paper determines F(s) exactly for s≤6, proving F(1)=1, F(2)=2, F(3)=4, F(4)=5, F(5)=6, and F(6)=8 for graphs whose complements have girth at least six.
• The paper reduces the general problem to a girth-six Ramsey extremal number and establishes c(s/log s)^(4/3)≤F(s)≤(1+o(1))s^(3/2)/log s, leaving the true asymptotic order open.
• The paper disproves Gyárfás’s conjecture by constructing almost perfect graphs with χ−ω arbitrarily large, using joins of distinct odd cycles while retaining χ-boundedness.

This paper by Kaiyang Lan and Wenlong Zhong resolves two open problems in the theory of $\chi$-boundedness. The first, due to Sivaraman (2606.29873), asks for the optimal $\chi$-bounding function for graphs whose complements have girth at least $6$; the authors determine this function exactly for clique numbers up to $6$ and give asymptotic bounds in general. The second, due to Gyárfás, asks whether "almost perfect" graphs are $\chi$-bounded by $x+1$; the authors answer negatively by constructing almost perfect graphs whose chromatic number exceeds their clique number by an arbitrarily large additive amount.

The Sivaraman problem and the function F(s)

A graph class is $\chi$-bounded if $\chi(H)$ can be bounded above by a function of $\omega(H)$ for every induced subgraph $H$. Sivaraman asked for the optimal bounding function for the class of graphs $\chi$0 with $\chi$1, i.e., graphs whose complements are $\chi$2-free. Since such graphs are $\chi$3-free, Wagon's classical bound $\chi$4 applies, giving a quadratic bound; the question is how much can be improved.

The paper defines

$\chi$5

and proves two main results about it. First, there exists a constant $\chi$6 such that

$\chi$7

Second, the exact values $\chi$8, $\chi$9, $6$0, $6$1, $6$2, and $6$3 hold, each sharp.

Reduction to a Ramsey-type extremal number

The key structural device is to pass to the complement: setting $6$4, the condition becomes that $6$5 is $6$6-free (girth at least six), and $6$7. The authors introduce the girth-six Ramsey extremal number

$6$8

and prove the reduction lemma

$6$9

The lower bound uses the elementary identity $6$0 for triangle-free $6$1: since $6$2, the complement of an extremal graph has chromatic number at least half its order. For the upper bound, a maximal matching leaves at most $6$3 uncovered vertices (they form a stable set), so $6$4. Consequently, whenever $6$5, one has $6$6, so determining $6$7 asymptotically reduces to determining $6$8.

Asymptotic bounds via second neighborhoods and Shearer's bound

The upper bound on $6$9 exploits the girth condition directly: in a graph of girth at least six, the second neighborhood $\chi$0 of any vertex is stable, and each of its vertices has a unique neighbor in $\chi$1 (otherwise a $\chi$2 arises). Summing over all vertices yields $\chi$3, and Cauchy's inequality gives average degree $\chi$4. Applying the Ajtai–Komlós–Szemerédi–Shearer independent set bound $\chi$5 for triangle-free graphs then forces $\chi$6, hence $\chi$7.

The lower bound comes from Spencer's short-cycle Ramsey construction, which gives $\chi$8, transferred to $\chi$9 through the reduction lemma. A notable feature of these bounds is the gap between the exponents $x+1$0 and $x+1$1; the true order of $x+1$2 remains undetermined, and closing this gap is left open.

Exact small values via Tutte–Berge

The exact values $x+1$3, $x+1$4, and $x+1$5 require controlling the matching number of the complement. The method combines three ingredients:

• Small Ramsey estimates: careful degree-counting arguments establish $x+1$6 (attained by $x+1$7), $x+1$8 (attained by $x+1$9), $\chi$0, and $\chi$1, along with uniqueness of the equality cases.
• The coloring–matching identity: $\chi$2 for triangle-free $\chi$3.
• The Tutte–Berge formula: writing $\chi$4, a finite case analysis over component-order partitions shows that if $\chi$5 exceeded the target ($\chi$6, $\chi$7, or $\chi$8 respectively), then $\chi$9 would exceed $\chi(H)$0, $\chi(H)$1, or $\chi(H)$2, contradicting $\chi(H)$3.

Sharpness is witnessed by complements of odd cycles: $\chi(H)$4 gives $\chi(H)$5 and $\chi(H)$6 gives $\chi(H)$7, while $\chi(H)$8 requires the disjoint union $\chi(H)$9, whose complement satisfies $\omega(H)$0 with $\omega(H)$1. The remaining small values follow more simply: $\omega(H)$2 from bipartiteness of triangle-free graphs on at most four vertices, and $\omega(H)$3 from Gaspers and Huang's theorem that $\omega(H)$4-free graphs are $\omega(H)$5-colorable, with sharpness from $\omega(H)$6.

An implication worth noting: the optimal $\omega(H)$7-bounding function for Sivaraman's class grows strictly subquadratically — indeed at most on the order of $\omega(H)$8 — improving Wagon's quadratic bound substantially, while remaining superlinear.

Almost perfect graphs: a negative answer to Gyárfás

Lovász characterized perfect graphs by the inequality $\omega(H)$9 for all induced subgraphs $H$0. Gyárfás proposed the relaxation obtained by weakening this to $H$1, calling such graphs almost perfect, and asked whether they satisfy $H$2. Scott and Seymour had shown almost perfect graphs are $H$3-bounded, but without the specific linear bound; Gyárfás confirmed the bound when $H$4 using Folkman's theorem.

The paper refutes the conjecture in a strong form. For each positive integer $H$5, define

$H$6

the join of the first $H$7 odd cycles of length at least five. The join formulas give $H$8 and $H$9, so $\chi$00.

The substantive part of the proof is verifying that $\chi$01 is almost perfect. Every induced subgraph of $\chi$02 is a join $\chi$03 where each $\chi$04 is an induced subgraph of $\chi$05. Proper induced subgraphs of an odd cycle are forests, satisfying $\chi$06, while full cycles contribute exactly one excess vertex. Writing $\chi$07 and letting $\chi$08 index the parts that are entire cycles, one needs

$\chi$09

Since the cycle lengths are distinct, the quantities $\chi$10 over $\chi$11 are distinct nonnegative integers, so $\chi$12 where $\chi$13; multiplying by $\chi$14 and adding $\chi$15 gives $\chi$16. This closes the induction-free verification that $\chi$17 for every induced subgraph $\chi$18.

Consequently, no constant $\chi$19 satisfies $\chi$20 for all almost perfect graphs $\chi$21: choosing $\chi$22 yields a counterexample. This decisively separates almost perfect graphs from perfect graphs with respect to additive $\chi$23-bounding, even though both classes are $\chi$24-bounded by Scott and Seymour's result.

Limitations and open questions

Two gaps remain open. First, the asymptotic behavior of $\chi$25 is pinned down only up to the exponent gap between $\chi$26 and $\chi$27; determining whether $\chi$28, or finding the correct exponent, is unresolved. Second, while the paper disproves the additive bound $\chi$29 for almost perfect graphs, it does not settle what the optimal $\chi$30-bounding function for this class is beyond the existence guaranteed by Scott and Seymour; in particular, whether some linear function $\chi$31 bounds the class remains open. The construction $\chi$32 itself has $\chi$33 but ratio $\chi$34, so it does not rule out multiplicative linear bounds.

Conclusion

The paper settles Sivaraman's problem for clique numbers up to six with sharp constants, establishes polynomial asymptotic bounds $\chi$35 via a clean complement reduction combined with second-neighborhood counting and Shearer's independence bound, and disproves Gyárfás's conjecture on almost perfect graphs through an explicit join-of-odd-cycles construction. Together, the results sharpen the picture of $\chi$36-boundedness near perfection: the complement-girth class admits much better than quadratic bounds, while the almost-perfect class, though $\chi$37-bounded, admits no bound within any constant additive term of the clique number.