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Abhishek Dhawan

Publications and source records attributed to Abhishek Dhawan.

At least 19 recordsLinked to original sources
arXiv2026

Random independent sets in uncrowded hypergraphs

Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies \[ χ_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}. \] In fact, we prove that this holds for $k$-uniform $d$-degenerate hypergraphs of girth at least $g$, for any given $g \ge 5$. As a corollary, we obtain improved bounds on the fractional chromatic number of $d$-degenerate linear hypergraphs. This work builds upon a recent result by Allen, Dhawan, and Noel, extending it from graphs to hypergraphs. In addition to overcoming the new difficulties that arise in the hypergraph setting, our approach yields a simpler proof even in the original graph case. Our proof of the upper bound uses a simpler iterative procedure for sampling independent sets. We also establish bounds for fractional colorings with local demands, a framework introduced by Kelly and Postle, verifying a recent conjecture of Yu and Zhang. As a consequence, we obtain a degree-sequence bound on the independence number of uncrowded hypergraphs with a leading constant matching the shattering threshold. For the matching lower bound, we use a hypergraph variant of the uniform attachment model and harmonic vertex weights to bound the fractional chromatic number via linear programming duality, then remove all short cycles by deleting vertices of negligible total weight. We also replace the Catalan-number argument used in the graph case with a matrix-norm estimate, simplifying the analysis.

math.CO↗
arXiv2026

Fractional coloring via entropy

In recent work, Martinsson and Steiner proved that triangle-free $d$-degenerate graphs have fractional chromatic number $χ_f(G) = O\left(\frac{d}{\log d}\right)$. Here, we introduce an alternate proof of the bound rooted in the analysis of the entropy of certain random variables. Beyond simplifying the original argument, our technique naturally generalizes to broader settings. In this paper, we focus on two extensions of the result. First, we consider locally $r$-colorable graphs $G$, where $χ(G[N(v)]) \le r$ for every vertex $v$. We show that $d$-degenerate locally $r$-colorable graphs satisfy $χ_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$, strengthening a result of Alon (1996) on the independence number of such graphs. Second, we extend Martinsson and Steiner's result to $r$-uniform $d$-degenerate hypergraphs $H$ with girth at least $4$, showing that $χ_f(H) \le c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$ for a constant $c_r > 0$. This yields a strict generalization of a seminal result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi (1982) on the independence number of uncrowded hypergraphs. Via random sampling as introduced by Duke, Lefmann, and Rödl (1995), we obtain an improved bound on the independence number of linear hypergraphs, establishing a recent conjecture of Verstraëte and Wilson (2026). Our arguments generalize to the setting of fractional colorings with local demands, introduced by Kelly and Postle (2024). This yields previously unknown degree-sequence lower bounds on the independence number across each setting considered. Furthermore, our approach is constructive, yielding efficient randomized algorithms for sampling independent sets in these contexts.

math.CO↗
arXiv2026

Fractional DP-colorings of $d$-degenerate locally sparse graphs

Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $χ_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $χ_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ χ_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $χ_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $χ_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $Δ$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.

math.CO↗
arXiv2026

List coloring uncrowded hypergraphs at the shattering threshold

Improving an earlier bound of Frieze and Mubayi, Iliopoulos showed that any $k$-uniform uncrowded hypergraph of maximum degree $Δ$ has list chromatic number at most $(1+o(1))(k-1)\Big(\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}$. Determining the optimal leading constant in this bound remains a major open problem. A natural target for the constant, suggested by the coupon-collector heuristic, is $(k-1)^{\frac{1}{k-1}}$, which also arises independently as the shattering threshold for coloring and finding large independent sets in sparse random hypergraphs. We show that uncrowded hypergraphs attain this threshold: for $k \ge 2$, every $k$-uniform uncrowded hypergraph $\mathcal{H}$ of maximum degree at most $Δ$ satisfies $$χ_\ell(\mathcal{H})\le(1+o(1))\Big((k-1)\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}.$$ This matches the coupon-collector heuristic, and establishes a 2019 conjecture of Molloy for uncrowded hypergraphs. As a consequence, we show that every $k$-uniform linear hypergraph of maximum degree at most $Δ$ has chromatic number at most $c_k\Big(\fracΔ{\log Δ}\Big)^{\frac{1}{k-1}}$ with $c_k=1-o_k(1)$, resolving, in a stronger form, a recent conjecture of Verstraëte and Wilson on the independence number of linear hypergraphs. Our techniques yield the first palette sparsification result for hypergraph coloring using $o(Δ^{\frac{1}{k-1}})$ colors per vertex. Specifically, for an uncrowded $k$-graph $\mathcal{H}$ and $q=Θ\Big({\Big(\fracΔ{γ\logΔ}\Big)^{\frac{1}{k-1}}}\Big)$, sampling $Θ(Δ^{\fracγ{k-1}})$ colors per vertex suffices to obtain a proper $q$-coloring of $\mathcal{H}$ w.h.p. As an algorithmic consequence, we obtain a single-pass streaming algorithm using $O\big(n^{1+o(1)}\big)$ space for uncrowded $k$-graphs, and, via hypergraph partitioning, for linear hypergraphs as well.

math.CO↗
arXiv2026

Sharp bounds for the fractional chromatic number of high-girth $d$-degenerate graphs

Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $χ_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our proof is constructive: it gives an efficient randomized algorithm that, with high probability, computes a fractional coloring of weight at most $(1 + o(1))\frac{d}{\log d}$ in such graphs. Furthermore, we establish their conjectured lower bound in a stronger form: for any constant $g \ge 4$, there exist $d$-degenerate graphs having girth at least $g$ with $χ_f(G) \ge (1 - o(1))\frac{d}{\log d}$. This lower bound is achieved by analyzing a random graph based on the uniform attachment model. Notably, our results reveal that this model lacks the typical computational complexity barriers found in Erdős-Rényi graphs, where there is a conjectured factor-$2$ algorithmic gap for this problem.

math.CO↗
arXiv2026

Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs

We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erdős-Rényi hypergraphs, and (ii) the more challenging problem of finding large $γ$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $γ\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i γ_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.

cs.DS↗
arXiv2026

The strong chromatic index of $K_{t,t}$-free graphs

A strong edge coloring of a graph $G$ is an edge coloring $ϕ\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $χ'_s(G)$, is the minimum number of colors needed for a strong edge coloring of $G$. Erdős and Nešetřil conjectured in 1985 that if $G$ has maximum degree $d$, then $χ'_s(G) \leq \frac 54 d^2$. Mahdian showed in 2000 that if $G$ is $C_4$-free, then $χ'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$, and he conjectured that the same upper bound holds for $K_{t,t}$-free graphs. In this paper, we prove this conjecture and improve upon it to show the following: every $K_{t,t}$-free graph $G$ of maximum degree $d$ satisfies $χ'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$. We employ a variant of the Rödl nibble method to prove this result. The key new ingredient in our adaptation of the method is an application of the Kővári-Sós-Turán theorem to show that $H := L(G)^2$ satisfies certain structural properties. These properties, in conjunction with a variant of Talagrand's inequality to handle exceptional outcomes, allow us to concentrate the sizes of certain vertex sets through the nibble, even when these vertex sets have order smaller than the maximum codegree of $H$. We encapsulate these structural properties into a more general statement on list coloring that we believe to be of independent interest. In light of the conjectured computational threshold for coloring random graphs arising in average-case complexity theory, we suspect that our result is best possible using this approach.

math.CO↗
arXiv2025

Choosability of multipartite hypergraphs

A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. We show that $k$-partite $k$-graphs of maximum degree $Δ$ are $q$-choosable for $q \geq \left(\frac{4}{5}(k-1 + o(1))Δ/\log Δ\right)^{1/(k-1)}$. Our proof yields an efficient randomized algorithm for finding such a coloring, which shows that the conjectured algorithmic barrier for coloring pseudorandom $k$-graphs does not apply to $k$-partite $k$-graphs.

math.CO↗
arXiv2025

Coloring locally sparse graphs

A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $Δ$ is $Δ^2/f$-locally-sparse, then $χ(G) = O\left(Δ/\log f\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph $F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the Rödl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $χ(G) = O\left( Δ/\log\left(Δk^{-1/|V(F)|}\right)\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $χ(G)$ when $G$ is $K_{1, t, t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the more general so-called ''color-degree'' setting.

math.CO↗
arXiv2025

Independent sets and colorings of $K_{t,t,t}$-free graphs

Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $Δ$ has chromatic number $O(Δ/ \log Δ)$. This was previously known only for almost bipartite graphs, that is, for subgraphs of $K_{1,t,t}$ (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where $F$ is almost bipartite. We prove this conjecture for all $3$-colorable graphs $F$, i.e. subgraphs of $K_{t,t,t}$, representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erdős, Komlós, and Szemerédi from 1981 asserts that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $Ω(n \log d / d)$. We prove this conjecture in a strong form for all 3-colorable graphs $F$. More precisely, we show that every $n$-vertex $K_{t,t,t}$-free graph of average degree $d$ contains an independent set of size at least $(1 - o(1)) n \log d / d$, matching Shearer's celebrated bound for triangle-free graphs (the case $t = 1$) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the Rödl nibble method for constructing independent sets with a Turán-type result on $K_{t,t,t}$-free graphs.

math.CO↗
arXiv2025

The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs

We study the algorithmic task of finding large independent sets in Erdos-Renyi $r$-uniform hypergraphs on $n$ vertices having average degree $d$. Krivelevich and Sudakov showed that the maximum independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$. We show that the class of low-degree polynomial algorithms can find independent sets of density $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$ but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining $r$-partite hypergraphs. A hypergraph $H=(V,E)$ is $r$-partite if there is a partition $V=V_1\cup\cdots\cup V_r$ such that each edge contains exactly one vertex from each set $V_i$. We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random $r$-partite hypergraphs with $n$ vertices in each partition and average degree $d$. We prove that the maximum balanced independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$ asymptotically. Furthermore, we prove an analogous low-degree computational threshold of $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$. Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.

cs.CC↗
arXiv2025

List colorings of $k$-partite $k$-graphs

A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. In this note, we consider list colorings for such hypergraphs. We show that for any $\varepsilon > 0$ if each vertex $v \in V(H)$ is assigned a list of size $|L(v)| \geq \left((k-1+\varepsilon)Δ/\log Δ\right)^{1/(k-1)}$, then $H$ admits a proper $L$-coloring, provided $Δ$ is sufficiently large. Up to a constant factor, this matches the bound on the chromatic number of simple $k$-graphs shown by Frieze and Mubayi, and that on the list chromatic number of triangle free $k$-graphs shown by Li and Postle. Our results hold in the more general setting of ``color-degree'' as has been considered for graphs. Furthermore, we establish a number of asymmetric statements matching results of Alon, Cambie, and Kang for bipartite graphs.

math.CO↗
arXiv2025

Sharp Online Hardness for Large Balanced Independent Sets

We study the algorithmic problem of finding large $γ$-balanced independent sets in dense random bipartite graphs; an independent set is $γ$-balanced if a $γ$ proportion of its vertices lie on one side of the bipartition. In the sparse regime, Perkins and Wang established tight bounds within the low-degree polynomial (LDP) framework, showing a factor-$1/(1-γ)$ statistical-computational gap via the Overlap Gap Property (OGP) framework tailored for stable algorithms. However, these techniques do not appear to extend to the dense setting. For the related large independent set problem in dense random graph, the best known algorithm is an online greedy procedure that is inherently unstable, and LDP algorithms are conjectured to fail even in the "easy" regime where greedy succeeds. We show that the largest $γ$-balanced independent set in dense random bipartite graphs has size $α:=\frac{\log_b n}{γ(1-γ)}$ whp, where $n$ is the size of each bipartition, $p$ is the edge probability, and $b=1/(1-p)$. We design an online algorithm that achieves $(1-ε)(1-γ)α$ whp for any $ε>0$. We complement this with a sharp lower bound, showing that no online algorithm can achieve $(1+ε)(1-γ)α$ with nonnegligible probability. Our results suggest that the same factor-$1/(1-γ)$ gap is also present in the dense setting, supporting its conjectured universality. While the classical greedy procedure on $G(n,p)$ is straightforward, our algorithm is more intricate: it proceeds in two stages, incorporating a stopping time and suitable truncation to ensure that $γ$-balancedness-a global constraint-is met despite operating with limited information. Our lower bound utilizes the OGP framework; we build on a recent refinement of this framework for online models and extend it to the bipartite setting.

cs.DS↗
arXiv2025

Toward Vu's conjecture

In 2002, Vu conjectured that graphs of maximum degree $Δ$ and maximum codegree at most $ζΔ$ have chromatic number at most $(ζ+o(1))Δ$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $ζ$ is close to $1$, by Hurley, de Verclos, and Kang. In this paper we provide the first progress in the sparse regime $ζ\ll 1$, the case of primary interest to Vu. We show that there exists $ζ_0 > 0$ such that for all $ζ\in [\log^{-32}Δ,ζ_0]$, the following holds: if $G$ is a graph with maximum degree $Δ$ and maximum codegree at most $ζΔ$, then $χ(G) \leq (ζ^{1/32} + o(1))Δ$. We derive this from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. Our more general result also extends to the list coloring setting, which is of independent interest.

math.CO↗
arXiv2025

Fast algorithms for Vizing's theorem on bounded degree graphs

Vizing's theorem states that every graph $G$ of maximum degree $Δ$ can be properly edge-colored using $Δ+ 1$ colors. The fastest currently known $(Δ+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. We investigate the case when $Δ$ is constant, i.e., $Δ= O(1)$. In this regime, the runtime of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. Prior to this work, no linear-time $(Δ+1)$-edge-coloring algorithm was known for any $Δ\geq 4$. Using some of the same ideas, we also develop new algorithms for $(Δ+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, when $Δ$ is constant, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. Although our focus is on the constant $Δ$ regime, our results remain interesting for $Δ$ up to $\log^{o(1)} n$, since the dependence of their running time on $Δ$ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.

cs.DS↗
arXiv2025

Bounds for the independence and chromatic numbers of locally sparse graphs

In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ cliques of size $r$. For $r \geq 3$ and $ε\in [0, 1]$, we show that an $n$-vertex $(Δ^{εr}, r)$-locally-sparse graph $G$ of maximum degree $Δ$ satisfies $α(G) \geq (1-o(1))\dfrac{n}{ηΔ}$ and $χ(G) \leq Θ\left(ηΔ\right)$, where $η:= ε+ \dfrac{r\log\log Δ}{\log Δ}$. For $ε$ not too large, the hidden constant in the $Θ(\cdot)$ can be taken to be $1+o(1)$. Setting $ε= 0$, we recover classical results on $K_{r+1}$-free graphs due to Shearer and Johansson, which were more recently improved by Davies, Kang, Pirot, and Sereni. We prove a stronger result on the independence number in terms of the occupancy fraction in the hard-core model, and establish a local version of the coloring result in the more general setting of correspondence coloring.

math.CO↗
arXiv2025

Coloring graphs with forbidden almost bipartite subgraphs

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $χ(G) \leq (c(F) + o(1)) Δ/ \logΔ$ whenever $G$ is an $F$-free graph of maximum degree $Δ$. The largest class of connected graphs $F$ for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph $K_{1,t,t}$ for some $t \in \mathbb{N}$). However, the optimal value for $c(F)$ remains unknown even for such graphs. Bollobás showed, using random regular graphs, that $c(F) \geq 1/2$ when $F$ contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of $c(K_{1,t,t}) \leq t$. We improve this to a uniform constant, showing $c(F) \leq 4$ for every almost bipartite graph $F$. This surprisingly makes the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results.

math.CO↗
arXiv2025

Balanced colorings of Erdős-Rényi hypergraphs

An $r$-uniform hypergraph $H = (V, E)$ is $r$-partite if there exists a partition of the vertex set into $r$ parts such that each edge contains exactly one vertex from each part. We say an independent set in such a hypergraph is balanced if it contains an equal number of vertices from each partition. The balanced chromatic number of $H$ is the minimum value $q$ such that $H$ admits a proper $q$-coloring where each color class is a balanced independent set. In this note, we determine the asymptotic behavior of the balanced chromatic number for sparse $r$-uniform $r$-partite Erdős--Rényi hypergraphs. A key step in our proof is to show that any balanced colorable hypergraph of average degree $d$ admits a proper balanced coloring with $r(r-1)d + 1$ colors. This extends a result of Feige and Kogan on bipartite graphs to this more general setting.

math.CO↗