The \textit{$r$-dynamic choosability} of a graph $G$, written ${\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$ has at least $\min\{d_G(v),r\}$ neighbors of distinct colors. Let ${\rm ch}(G)$ denote the choice number of $G$. In this paper, we prove ${\rm ch}_r(G)\leq (1+o(1)){\rm ch}(G)$ when $\frac{Δ(G)}{δ(G)}$ is bounded. We also show that there exists a constant $C$ such that for the random graph $G=G(n,p)$ with $\frac{2}{n}<p\leq \frac{1}{2}$, it holds that ${\rm ch}_2(G)\leq {\rm ch}(G) + C$, asymptotically almost surely. Also if $G$ is triangle-free regualr graph, then ${\rm ch}_2(G)\leq {\rm ch}(G)+86$ holds.
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=Ω(\sqrt n)$, and at most $\min\bigl(2n-Ω(\sqrt n),2n-D-2\bigr)$ edges.
This paper has been withdrawn by the author. Peterson and Woodall previously proved that the list-edge-colouring conjecture holds for graphs without odd cycles of length 5 or longer. D. Peterson and D. R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Mathematics 202 (1999), 191-199. D. Peterson and D. R. Woodall, Erratum to "Edge-choosability in line-perfect multigraphs", Discrete Mathematics 260 (2003), 323-326.
Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…
A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for every $(\ell, k)$-list assignment $L$. We prove that if $G$ is a planar graph with $(4,2)$-list assignment $L$ and for every triangle $T \subseteq G$ we have that $|\bigcap_{v \in V(T)} L(v)| \neq 2$, then $G$ is $L$-colourable. In fact, we prove a slightly stronger result: if $G$ contains a clique $H$ such that $V(H) \cap V(T) \neq \emptyset$ for every triangl…
We study choosability with separation which is a constrained version of list coloring of graphs. A (k,d)-list assignment L on a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3,1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3,1)-choosable. In addition…
We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $Δ(G)$ satisfies $\lcl(G)\ge \ceil{Δ(G)/2}+1$. In this paper, we prove the following results: (1) if $\mad(G)<12/5$ and $Δ(G)\ge 3$, then $\lcl(G)=\ceil{Δ(G)/2}+1$, and we give an infinite family of examples to show that this result is best possible; (2) if $\mad(G)<3$ and $Δ(G)\ge 9$, then $\lcl(G)\le\ceil{Δ(G)/2}+2$, and we give an infinite…
A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous to the problem of Havel regarding 3-colorability of planar graphs with triangles far apart.
We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For exam…
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thoma…
An (improper) graph colouring has "defect" $d$ if each monochromatic subgraph has maximum degree at most $d$, and has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than $\frac{2d+2}{d+2} k$ is $k$-choosable with defect $d$. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree $m$, no $(1-ε)m$ bound on the number …
Degree-choosability of proper conflict-free list coloring of sparse graphs
Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Faculty of Information Studies, Novo Mesto（新梅斯托信息学院）；Rudolfovo - Science and Technology Centre Novo Mesto（鲁多尔福沃-新梅斯托科学与技术中心）；School of Mathematical and Science, Zhejiang Normal University（浙江师范大学数理学院）
The triangle packing number $ν(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2ν(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random graph $G=G(n,m)$, when $m \le 0.2403n^{3/2}$ or $m\ge 2.1243n^{3/2}$. This is done by analyzing a greedy algorithm for finding large triangle packings in random graphs.
One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish the new bound $\frac{27}{1024}n^2$ on the number of edges in general case. We completely prove the conjecture for graphs of girth $\geq 5$, for graphs with independence number $\geq 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph.
Erdős conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$ for some absolute $\varepsilon >0$. Moreover, we show that the c…
We describe a general approach of determining the distribution of spanning subgraphs in the random graph $\G(n,p)$. In particular, we determine the distribution of spanning subgraphs of certain given degree sequences, which is a generalisation of the $d$-factors, of spanning triangle-free subgraphs, of (directed) Hamilton cycles and of spanning subgraphs that are isomorphic to a collection of vertex disjoint (directed) triangles.
We compute a minimum degree threshold sufficient for 3-partite graphs to admit a fractional triangle decomposition. Together with recent work of Barber, Kühn, Lo, Osthus and Taylor, this leads to bounds for exact decompositions and in particular the completion problem for sparse partial latin squares. Some extensions are considered as well.
We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$λ$ satisfying $λ\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [\emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica \textbf{24} (2004), pp.~403--426] that $(n,d,λ)$-graphs with $n\in 3\mathbb{N}$ and $λ\leq cd^{2}/n$ for a suitably small absolute constant~$c>0$ con…
In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if $p(n) \gg (\log n / n)^{1/2}$, then asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree at least $(2/3 + o(1))np$ contains a triangle packing that covers all but at most $O(p^{-2})$ vertices. Moreover, the assumption on $p$ is optimal up to the $(\log n)^{1/2}$ factor and the presence of the set of $O(p^{-2})$ uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which…
We prove that every triangle-free $4$-critical graph $G$ satisfies $e(G) \geq \frac{5v(G)+2}{3}$. This result gives a unified proof that triangle-free planar graphs are $3$-colourable, and that graphs of girth at least five which embed in either the projective plane, torus, or Klein Bottle are $3$-colourable, which are results of Grötzsch, Thomassen, and Thomas and Walls. Our result is nearly best possible, as Davies has constructed triangle-free $4$-critical graphs $G$ such that $e(G) = \frac{5v(G) + 4}{3}$. To prove this result, we prove a more general result characterizing sparse $4$-critic…
Local large deviations for triangles in sparse random graphs
Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）；Mount Holyoke College, Department of Mathematics and Statistics（芒特霍利奥克学院数学与统计系）
We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Γ$ the generalized Turán density $π_F(Γ)$ denotes the density of a maximum subgraph of $Γ$, which contains no copy of~$F$. Extending classical Turán type results for odd cycles, we show that $π_{F}(Γ)=1/2$ provided $F$ is an odd cycle and $Γ$ is a sufficiently pseudorandom graph. In particular, for $(n,d,λ)$-graphs $Γ$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-λ,λ]$, our result holds for odd cycles of length $\ell$, provided \[ λ^{\ell-2}\ll \frac{d^{\ell-1}}n…
The Turán number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^α)$ where $α> 1$, then any $n$-vertex graph with the property that each edge lies in exactly one copy of $H$ has $o(n^α)$ edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when $H$ is a non-bipartite graph, in particular when $H$ is a triangle, due to Ruzsa and Szemerédi. Using Sidon sets we exhib…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Iowa State University（爱荷华州立大学）；McGill University（麦吉尔大学）；Czech Technical University in Prague（布拉格捷克理工大学）
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $G_{n,p}$ for $p \geq C(\log n/n)^{1/Δ}$, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph $H$ with $Δ(H) \leq Δ$. However, this is typically only optimal when $Δ\in \{2,3\}$ and $H$ either contains a triangle ($Δ= 2$) or many copi…
Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $α$ and $β$. An $L$-recoloring sequence, starting from $α$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $α$ to $β$ if its initial coloring is $α$ and its final coloring is $β$. We prove there exists an $L$-recoloring sequence that transforms $α$ to $β$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G)<17/5$ and $L$ is a 6-assig…
What is the probability that the number of triangles in $\mathcal{G}_{n,p}$, the Erdős-Rényi random graph with edge density $p$, is at least twice its mean? Writing it as $\exp[- r(n,p)]$, already the order of the rate function $r(n,p)$ was a longstanding open problem when $p=o(1)$, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that $r(n,p)\asymp n^2p^2 \log (1/p)$ for $p \gtrsim \frac{\log n}n$; the exact asymptotics of $r(n,p)$ remained unknown. The following variational problem can be related to this large deviation question at $p\gtrsim \frac{\log …
We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $Γ$ be an $(n,d,λ)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-λ,λ]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $λ^{2k-1}\ll d^{2k}/n$, every subgraph $G$ of $Γ$ with $(1/2+o(1))e(Γ)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, Hàn, and the third author proved a weaker statement by allowing an extra polylogarithmic factor in the assumption $λ^{2k-1}\ll d^{2k}/n$, but we completely remove it and hence …
The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ …
A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of $K_n$ is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large $p$, every maximum triangle-free subgraph of $G(n,p)$ is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for $p > K \sqrt{\log n/n}$ for some constant $K$, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by $F_5$, which sometimes called as the generalized triangle, the 3-uniform hyper…
A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets of vertices, of sizes at least $sn^s$ and $sn$, anticomplete to each other. We prove this holds for a large class of graphs $H$, and we prove that something like it holds for all graphs $H$. Say $H$ is "almost-bipartite" if $H$ is triangle-free and $V(H)$ can be partitioned into a stable set and a set ind…
Given integers $n\ge Δ\ge 2$, let $\mathcal{T}(n, Δ)$ be the collection of all $n$-vertex trees with maximum degree at most $Δ$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible spectral gap condition forcing an $(n, d,λ)$-graph to be $\mathcal{T}(n, Δ)$-universal, namely, it contains all members of $\mathcal{T}(n, Δ)$ as a subgraph simultaneously. In this paper we show that for sufficiently large integer $n$ and all $Δ\in \mathbb{N}$, every $(n, d,λ)$-graph with \[ λ\le\frac{d}{2Δ^{5\sqrt{\log n}}} \] is $\mathcal{T}(n, Δ)$-universal. As an immediate …
A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. The maximum average degree of a graph $G$, denoted by $\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\mad(G)<4$, then $χ'_a(G)\leq{Δ(G)+2}$; if $\mad(G)<3$, then $χ'_a(G)\leq{Δ(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(Δ(G)+2)$-colorable.
We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every $n$-vertex graph with no 5-cycle can be made triangle-free by deleting $o(n^{3/2})$ edges. * For $r \geq 3$, every $n$-vertex $r$-graph with girth greater than $5$ has $o(n^{3/2})$ edges. * Every subset of $[n]$ without a nontrivial solution to the equation $x_1 + x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$.
We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph $G_0$ in the common random graph models ${\cal G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of $G_0$. This extends an argument given earlier by the second author for $G_0=K_3$ with a more restricted range of average degree. For all strictly balanced subgraphs $G_0$, our results gives much information on the distribution of the number of copies of $G_0$ that are not in large "cl…
Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree distribution) with efficiently calculable ground-truth solutions to the desired output. Reproducibility for current generators used in benchmarking are somewhat lacking in this respect due to their randomness: the output of a desired graph analytic can only be compared to expected values and not exact ground truth. Nonstochastic Kronecker product graphs meet these des…
How can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first $k$ moments of the graph spectrum (equivalent to the numbers of closed $k$-walks) appropriately normalized. We introduce a simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The Random Overlapping Communities (ROC) model is specified by a distribution on pairs $(s,q)$, $s \in \mathbb{Z}_+, q \in (0,1]$. A graph on $n$ vertices with average degree $d$ is gene…
