arXiv subjects
Publications and source records attributed to Tung H. Nguyen.
Resolving a 1985 open problem of Gyárfás, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erdős-Hajnal result for $P_5$.
Let $T$ be a forest. We study polynomially high-chromatic pure pairs in graphs with no $T$ as an induced subgraph ($T$-free graphs in other words), with applications to the polynomial Gyárfás-Sumner conjecture. In addition to reproving several known results in the literature, we deduce: $\bullet$ If $T=P_5$ is the five-vertex path, then every $T$-free graph $G$ with clique number $w\ge2$ contains a complete pair $(A,B)$ of induced subgraphs with $χ(A)\ge w^{-d}χ(G)$ and $χ(B)\ge 2^{-d}χ(G)$, for some universal $d\ge1$. The proof uses the recent Erdős-Hajnal result for $P_5$-free graphs. Via the classical Gyárfás path argument, such a ``polynomial versus linear high-$χ$ complete pairs'' result can be viewed as further supporting evidence for the polynomial Gyárfás-Sumner conjecture for $P_5$. In particular, it implies \[χ(G)\le w^{O(\log w/\log\log w)}\] which asymptotically improves the bound $χ(G)\le w^{\log w}$ of Scott, Seymour, and Spirkl. $\bullet$ If $T$ and a broom satisfy the polynomial Gyárfás-Sumner conjecture, then so does their disjoint union. Unifying earlier results of Chudnovsky, Scott, Seymour, and Spirkl, and of Scott, Seymour, and Spirkl, this gives new instances of $T$ for which the conjecture holds.
We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has at most $α(G)ω(G)^d$ vertices. This ``off-diagonal Ramsey'' statement implies that every such graph $G$ has fractional chromatic number at most $ω(G)^d$, and is another step towards the polynomial Gyárfás-Sumner conjecture for $P_5$. The proof uses the recent Erdős-Hajnal result for $P_5$ and adapts a decomposition argument for $P_5$-free graphs developed by the author in an earlier paper.
Fix $\varepsilon>0$ and a nonnull graph $H$. A well-known theorem of Rödl from the 80s says that every graph $G$ with no induced copy of $H$ contains a linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces a subgraph with maximum degree at most $\varepsilon\vert S\vert$ in $G$ or its complement. There are two extensions of this result: $\bullet$ quantitatively, Nikiforov (and later Fox and Sudakov) relaxed the condition "no induced copy of $H$" into "at most $κ\vert G\vert^{\vert H\vert}$ induced copies of $H$ for some $κ>0$ depending on $H$ and $\varepsilon$"; and $\bullet$ qualitatively, Chudnovsky, Scott, Seymour, and Spirkl recently showed that there exists $N>0$ depending on $H$ and $\varepsilon$ such that $G$ is $(N,\varepsilon)$-restricted, which means $V(G)$ has a partition into at most $N$ subsets that are $\varepsilon$-restricted. A natural common generalization of these two asserts that every graph $G$ with at most $κ\vert G\vert^{\vert H\vert}$ induced copies of $H$ is $(N,\varepsilon)$-restricted for some $κ,N>0$ depending on $H$ and $\varepsilon$. This is unfortunately false, but we prove that for every $\varepsilon>0$, $κ$ and $N$ still exist so that for every $d\ge0$, every graph with at most $κd^{\vert H\vert}$ induced copies of $H$ has an $(N,\varepsilon)$-restricted induced subgraph on at least $\vert G\vert-d$ vertices. This unifies the two aforementioned theorems, and is optimal up to $κ$ and $N$ for every value of $d$.
For integers $k\ge1$ and $m\ge2$, let $g(k,m)$ be the least integer $n\ge1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. Refining the recent result Girão and Narayanan that $g(k-1,k)\le 7k+1$ for all $k\ge2$, we prove that $g(k,m)\le \max(m+2k-2,\lceil(3+\frac{1}{16})k\rceil)$ for all $k\ge1$ and $m\ge2$. This sharpens earlier results of Alon, Kleitman, Saks, Seymour, and Thomassen, of Chudnovsky, Penev, Scott, and Trotignon, and of Penev, Thomassé, and Trotignon. Our result implies that $g(k,k+1)\le\lceil(3+\frac{1}{16})k\rceil$ for all $k\ge1$, making a step closer towards a conjecture of Thomassen from 1983 that $g(k,k+1)\le 3k+1$, which was originally a result with a false proof and was the starting point of this research area.
It is proved that for every $\varepsilon>0$, there exists $K>0$ such that for every integer $t\ge2$, every graph with chromatic number at least $Kt$ contains a minor with $t$ vertices and edge density at least $1-\varepsilon$. Indeed, building on recent work of Delcourt and Postle on linear Hadwiger's conjecture, for $\varepsilon\in(0,\frac{1}{256})$ we can take $K=C\log\log(1/\varepsilon)$ where $C>0$ is a universal constant, which extends their recent $O(t\log\log t)$ bound on the chromatic number of graphs with no $K_t$ minor.
Given a bipartite graph with bipartition $(A,B)$ where $B$ is equipartitioned into $k\ge2$ blocks, can the vertices in $A$ be picked one by one so that at every step, the picked vertices cover roughly the same number of vertices in each of these blocks? We show that, if each block has cardinality $m$, the vertices in $B$ have the same degree, and each vertex in $A$ has at most $cm$ neighbors in every block where $c>0$ is a small constant, then there is an ordering $v_1,\ldots,v_n$ of the vertices in $A$ such that for every $j\in\{1,\ldots,n\}$, the numbers of vertices with a neighbor in $\{v_1,\ldots,v_j\}$ in every two blocks differ by at most $\sqrt{2(k-1)c}\cdot m$. This is related to a well-known lemma of Steinitz, and partially answers an unpublished question of Scott and Seymour.