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A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$. Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp. One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.

If $L$ is a list assignment of $r$ colors to each vertex of an $n$-vertex graph $G$, then an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/r\rceil$ times. A graph is equitably $r$-choosable if it has an equitable $L$-coloring for every $r$-list assignment $L$. In 2003, Kostochka, Pelsmajer and West (KPW) conjectured that an analog of the famous Hajnal-Szemer\'edi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer $r$ every graph $G$ with maximum degree at most $r-1$ is equitably $r$-choosable. The main result of this paper is that for each $r\geq 9$ and each planar graph $G$, a stronger statement holds: if the maximum degree of $G$ is at most $r$, then $G$ is equitably $r$-choosable. In fact, we prove the result for a broader class of graphs -- the class ${\mathcal{B}}$ of the graphs in which each bipartite subgraph $B$ with $|V(B)|\ge3$ has at most $2|V(B)|-4$ edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in ${\mathcal{B}}$, in particular, for all planar graphs.

Previously, Erd\H{o}s, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the F\"{u}redi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0

A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$., Comment: 28 pages, 12 figures

The generalized coloring numbers col_r(G) (also denoted by scol_r(G)) and wcol_r(G) of a graph G were introduced by Kierstead and Yang as a generalization of the usual coloring number, and have found important theoretical and algorithmic applications. For each distance r, these numbers are determined by an "optimal" ordering of the vertices of G. We study the question of whether it is possible to find a single "uniform" ordering that is "good" for all distances r. We show that the answer to this question is essentially "yes". Our results give new characterizations of graph classes with bounded expansion and nowhere dense graph classes., Comment: 17 pages

The weak $r$-coloring numbers $wcol_r(G)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $col(G)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs. Let $G^p$ denote the $p$-th power of $G$. We show that, all integers $p >0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta(G) \leq \Delta$ satisfy $col(G^p) \in O(p \cdot wcol_{\lceil p/2\rceil}(G)(\Delta-1)^{\lfloor p/2\rfloor})$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2 < mad(G) \leq 2k$, then $ col(G^2) \leq (2k-1)\Delta(G)+2k+1$., Comment: 9 pages

For any graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$, which has an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$ in $G$. For odd $p$, Ne\v{s}et\v{r}il and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of $G^{[\natural p]}$ is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Ne\v{s}et\v{r}il and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph $G$ and odd positive integer $p$, the chromatic number of $G^{[\natural p]}$ is bounded by the weak $(2p-1)$-colouring number of $G$. For even $p$, we prove that $\chi(G^{[\natural p]})$ is at most the weak $(2p)$-colouring number times the maximum degree. For odd $p$, the existing lower bound on the number of colours needed to colour $G^{[\natural p]}$ when $G$ is planar is improved. Similar lower bounds are given for $K_t$-minor free graphs., Comment: 21 pages, 3 figures; error in proof corrected plus some minor changes

In 1963, Corr\'adi and Hajnal proved that for all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq2k$ contains $k$ disjoint cycles. The bound $\delta(G) \geq 2k$ is sharp. Here we characterize those graphs with $\delta(G)\geq2k-1$ that contain $k$ disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of $(2k-1)$-connected graphs with no $k$ disjoint cycles. Enomoto and Wang refined the Corr\'adi-Hajnal Theorem, proving the following Ore-type version: For all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices contains $k$ disjoint cycles, provided that $d(x)+d(y)\geq 4k-1$ for all distinct nonadjacent vertices $x,y$. We refine this further for $k\geq3$ and $n\geq3k+1$: If $G$ is a graph on $n$ vertices such that $d(x)+d(y)\geq 4k-3$ for all distinct nonadjacent vertices $x,y$, then $G$ has $k$ vertex-disjoint cycles if and only if the independence number $\alpha(G)\leq n-2k$ and $G$ is not one of two small exceptions in the case $k=3$. We also show how the case $k=2$ follows from Lov\'asz' characterization of multigraphs with no two disjoint cycles., Comment: 23 pages, 8 figures

First-fit is the online graph coloring algorithm that considers vertices one at a time in some order and assigns each vertex the least positive integer not used already on a neighbor. The maximum number of colors used by first-fit on graph G over all vertex orders is denoted \chi_{FF}(G). The exact value of R := \sup_G [\chi_{FF}(G) / \omega(G)] over interval graphs G is unknown. Pemmaraju, Raman, and Varadarajan (2004) proved R <= 10, and this can be improved to 8. Witsenhausen (1976) and Chrobak and \'Slusarek (1988) showed R >= 4, and \'Slusarek (1993) improved this to 4.45. We prove R >= 5., Comment: Accepted to the European Journal of Combinatorics

Let $\mathrm{ch}(G)$ denote the choice number of a graph $G$, and let $K_{s*k}$ be the complete $k$-partite graph with $s$ vertices in each part. Erd\H{o}s, Rubin, and Taylor showed that $\mathrm{ch}( K_{2*k})=k$, and suggested the problem of determining the choice number of $K_{s*k}.$ The first author established $\mathrm{ch}( K_{3*k})=\left\lceil \frac{4k-1}{3}\right\rceil$. Here we prove $\mathrm{ch} (K_{4*k})=\left\lceil \frac{3k-1}{2}\right\rceil$.

In 1963, Corr\'adi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k-1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree $\delta(G) \ge 2k-1$ that do not contain k disjoint cycles. We use this result to answer Dirac's question in full., Comment: 7 pages, 2 figures. To appear in Combinatorica

For $k \in \mathbb N$, Corr\'adi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $\delta(G) \ge 2k$ has a $C_3$-factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle $C_3$. Wang proved that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $\delta_t(\overrightarrow G):=\min_{v\in V}(deg^-(v)+deg^+(v)) \ge 3(3k-1)/2$ has a $\overrightarrow C_3$-factor, where $\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers $a \ge 1$ and $b \ge 0$ with $a+b=k$, every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $\delta_t(\overrightarrow G)\ge 4k-1$ has a factor consisting of $a$ copies of $\overrightarrow T_3$ and $b$ copies of $\overrightarrow C_3$, where $\overrightarrow T_3$ is the transitive tournament on three vertices. In particular, using $b=0$, there is a $\overrightarrow T_3$-factor of $\overrightarrow G $, and using $a=1$, it is possible to obtain a $\overrightarrow C_3$-factor of $\overrightarrow G$ by reversing just one edge of $\overrightarrow G$. All these results are phrased and proved more generally in terms of undirected multigraphs. We conjecture that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum semidegree $\delta_0(\overrightarrow G):=\min_{v\in V}\min(deg^-(v),deg^+(v)) \ge 2k$ has a $\overrightarrow C_3$-factor, and prove that this is asymptotically correct., Comment: 19 pages, 1 figure

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result., Comment: 16 pages, 3 figures

In 1962 P\'osa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of P\'osa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 2n/3. Still in 1996, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved P\'osa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n > n_0, where n_0 is a very large constant. Here we show without using these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the recent work of Levitt, Szemer\'edi and S\'ark\"ozy, but our methods are based on techniques that were available in the 90's., Comment: 20 pages, to appear in Random Structures & Algorithms