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Kaiyang Lan

Publications and source records attributed to Kaiyang Lan.

9 recordsLinked to original sources
arXiv2026

An optimal $χ$-bound for $\{P_7, \text{diamond}\}$-free graphs

A $\textit{diamond}$ is a graph obtained by removing an edge from the complete graph on four vertices. A graph is $\{P_7, \text{diamond}\}$-$\textit{free}$ if it contains no induced subgraph isomorphic to a seven-vertex path or a diamond. Schiermeyer (2020) asked whether there is a constant $c$ such that $χ(G)\leqω(G)+c$ for every $\{P_7, \text{diamond}\}$-free graph $G$. We answer this question affirmatively by proving $χ(G) \leq ω(G)+3$. Moreover, the bound is optimal.

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arXiv2026

The chromatic number of $\{\mathrm{ISK}_4, \textup{diamond}\}$-free graphs

An $\mathrm{ISK}_4$ is a graph that is isomorphic to a subdivision of $K_4$; a \textit{diamond} is $K_4$ minus one edge; and a \textit{bowtie} consists of two triangles sharing exactly one vertex. Lévêque, Maffray, and Trotignon~(2012) conjectured that every $\mathrm{ISK}_4$-free graph is $4$-colorable. Chen, Chen, Cui, Feng, and Liu~(2021) proved this for $\{\textup{diamond},\textup{bowtie}\}$-free graphs. We prove this conjecture for diamond-free graphs. Moreover, the bound is optimal.

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arXiv2026

An improved polynomial $χ$-bound for $\{P_5,C_5\}$-free graphs

Nguyen~\cite{Nguyen2025} recently proved that every $\{P_5,C_5\}$-free graph $G$ satisfies $χ(G)\leq ω(G)^{40}$. Building on his framework, we introduce two refinements, namely a sharper cutset decomposition using the $C_5$-free condition and an improved density-increment argument. These yield a polynomial $χ$-binding function with exponent $24$, improving the previous bound of $40$.

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arXiv2026

The optimal $χ$-bound for $\{P_6, \text{dart}, K_4\}$-free graphs

A \textit{diamond} is a graph obtained from \(K_4\) by removing an edge, and a \textit{dart} is a graph obtained from a diamond by adding a pendant edge to a vertex of degree 3. We prove that every $\{P_6, \text{dart}, K_4\}$-free graph is 6-colorable. This improves the previous bound of 7 due to Hong and Xu \cite{HongXu2025} and resolves their open question on the optimality of the bound. Our result also extends a theorem of Karthick and Mishra~\cite{KarthickMishra2018}, who proved 6-colorability for the class of \(\{P_6, \text{diamond}, K_4\}\)-free graphs.

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arXiv2026

On a problem of Sivaraman and a problem of Gyárfás

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 $χ(G)\le ω(G)+c$.

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arXiv2026

A non-hereditary Pollyanna class that is not strongly Pollyanna

Chudnovsky, Cook, Davies, and Oum introduced the notion of Pollyanna graph classes: a class $\mathcal{C}$ is Pollyanna if for every $χ$-bounded class $\mathcal{F}$, the intersection $\mathcal{C} \cap \mathcal{F}$ is polynomially $χ$-bounded. They further defined $\mathcal{C}$ to be strongly Pollyanna if it is $k$-strongly Pollyanna for some integer $k$, meaning that $\mathcal{C} \cap \mathcal{F}$ is polynomially $χ$-bounded for every $k$-good class $\mathcal{F}$. They asked whether there are Pollyanna graph classes that are not strongly Pollyanna. In this note we answer this question affirmatively, under the literal interpretation that graph classes are not required to be hereditary. We construct a class $\mathcal{C}$ that is Pollyanna but, for every $k \ge 1$, is not $k$-strongly Pollyanna; in particular $\mathcal{C}$ is not strongly Pollyanna.

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arXiv2026

On two conjectures of Hoàng

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $ω(B) < ω(H)$. It is easy to see that the chromatic number of a perfectly divisible graph is at most $\binom{ω(G)+1}{2}$. Hoàng conjectured that every graph $G$ with $α(G) \le 3$ is perfectly divisible. We disprove this conjecture. In the same vein, a graph $G$ with at least one edge is $k$-divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into $k$ sets, none of which contains a largest clique of $H$. It is easy to see that the chromatic number of a $k$-divisible graph is at most $k^{ω-1}$. Hoàng conjectured that every even-hole-free graph is 3-divisible. We confirm this conjecture.

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arXiv2025

Trisimplicial vertices in (fork, odd parachute)-free graphs

An {\em odd hole} in a graph is an induced subgraph which is a cycle of odd length at least five. An {\em odd parachute} is a graph obtained from an odd hole $H$ by adding a new edge $uv$ such that $x$ is adjacent to $u$ but not to $v$ for each $x\in V(H)$. A graph $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A vertex of a graph is {\em trisimplicial} if its neighbourhood is the union of three cliques. In this paper, we prove that $χ(G)\leq \binom{ω(G)+1}{2}$ if $G$ is a (fork, odd parachute)-free graph by showing that $G$ contains a trisimplicial vertex when $G$ is nonperfectly divisible. This generalizes some results of Karthick, Kaufmann and Sivaraman [{\em Electron. J. Combin.} \textbf{29} (2022) \#P3.19], and Wu and Xu [{\em Discrete Math.} \textbf{347} (2024) 114121]. As a corollary, every nonperfectly divisible claw-free graph contains a trisimplicial vertex.

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