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Luke Postle

Publications and source records attributed to Luke Postle.

At least 73 records · Page 4Linked to original sources
arXiv2018

Bounding $χ$ by a fraction of $Δ$ for graphs without large cliques

The greedy coloring algorithm shows that a graph of maximum degree at most $Δ$ has chromatic number at most $Δ+ 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $Δ\geq 3$ and the graph contains no clique of size $Δ+ 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $Δ+ 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $Ω(\ln(Δ)^{-1})$ or $Ω\left(\frac{\ln(\ln(Δ))}{\ln(Δ)}\right)$ for graphs with no triangles or no cliques of any fixed size, respectively. Notably missing is a $\textit{linear}$ improvement on the "greedy bound" for graphs without large cliques. In this paper, we prove that for sufficiently large $Δ$, if $G$ is a graph with maximum degree at most $Δ$ and no clique of size $ω$, then $$χ(G) \leq 72Δ\sqrt{\frac{\ln(ω)}{\ln(Δ)}}.$$ This implies that for sufficiently large $Δ$, if $ω^{(72c)^2} \leq Δ$ then $χ(G) \leq Δ/c$. This bound actually holds for the list-chromatic and even the correspondence-chromatic number (also known as the DP-chromatic number). In fact, we prove what we call a "local version" of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. Our result simultaneously implies the linear improvement over the "greedy bound" and the two aforementioned results of Johansson.

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arXiv2017

The List Linear Arboricity of Graphs

A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating that every $d$-regular graph $G$ has linear arboricity $\lceil \frac{d+1}{2} \rceil$. In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC was raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Arboricity Conjecture holds asymptotically.

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arXiv2017

3 List Coloring Graphs of Girth at least Five on Surfaces

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable. In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvorak's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has $2^{Ω(n)-O(g)}$ distinct 3-list-colorings.

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arXiv2017

On the Minimum Number of Edges in Triangle-Free 5-Critical Graphs

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical graph, then |E(G)| >= (9/4 + e)|V(G)|- 5/4 - dT(G), where T(G) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.

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arXiv2017

On the clique number of the square of a line graph and its relation to Ore-degree

In 1985, Erdős and Nešetřil conjectured that the square of the line graph of a graph $G$, that is $L(G)^2$, can be colored with $\frac{5}{4}Δ(G)^2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $ω(L(G)^2)$, is at most $\frac{5}{4}Δ(G)^2$. In 2015, Śleszyńska-Nowak proved that $ω(L(G)^2)\le \frac{3}{2}Δ(G)^2$. In this paper, we prove that $ω(L(G)^2)\le \frac{4}{3}Δ(G)^2$. This theorem follows from our stronger result that $ω(L(G)^2)\le \frac{σ(G)^2}{3}$ where $σ(G) := \max_{uv\in E(G)} d(u) + d(v)$, is the Ore-degree of the graph $G$.

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arXiv2016

Chromatic index determined by fractional chromatic index

Given a graph $G$ possibly with multiple edges but no loops, denote by $Δ$ the {\it maximum degree}, $μ$ the {\it multiplicity}, $χ'$ the {\it chromatic index} and $χ_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $Δ\le χ_f' \le χ' \le Δ+ μ$, where the upper bound is a classic result of Vizing. While deciding the exact value of $χ'$ is a classic NP-complete problem, the computing of $χ_f'$ is in polynomial time. In fact, it is shown that if $χ_f' > Δ$ then $χ_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that $χ'=\lceilχ_f'\rceil$ if $χ'\ge Δ+2$, which is commonly referred as Goldberg's conjecture. In this paper, we show that if $χ' >Δ+\sqrt[3]{Δ/2}$ then $χ'=\lceilχ_f'\rceil$. The previous best known result is for graphs with $χ'> Δ+\sqrt{Δ/2}$ obtained by Scheide, and by Chen, Yu and Zang, independently. It has been shown that Goldberg's conjecture is equivalent to the following conjecture of Jakobsen: {\it For any positive integer $m$ with $m\ge 3$, every graph $G$ with $χ'>\frac{m}{m-1}Δ+\frac{m-3}{m-1}$ satisfies $χ'=\lceilχ_f'\rceil$.} Jakobsen's conjecture has been verified for $m$ up to 15 by various researchers in the last four decades. We show that it is true for $m\le 23$. Moreover, we show that Goldberg's conjecture holds for graphs $G$ with $Δ\leq 23$ or $|V(G)|\leq 23$.

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arXiv2016

Five-list-coloring graphs on surfaces III. One list of size one and one list of size two

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $ϕ$ of $G$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\ge2$. Here we characterize when an $L$-coloring exists if $|L(v_1)|\ge1$ and $|L(v_2)|\ge2$. This result is a lemma toward a more general theorem along the same lines, which we will use to prove that minimally non-$L$-colorable planar graphs with two precolored cycles of bounded length are of bounded size. The latter result has a number of applications which we pursue elsewhere.

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arXiv2016

Bounded Diameter Arboricity

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose components has diameter at most $d$. A class of graphs has bounded diameter arboricity $k$ if there exists a natural number $d$ such that every graph in the class has diameter-$d$ arboricity at most $k$. We conjecture that the class of graphs with arboricity at most $k$ has bounded diameter arboricity at most $k+1$. We prove this conjecture for $k\in \{2,3\}$ by proving the stronger assertion that the union of a forest and a star forest can be partitioned into two forests of diameter at most 18. We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least $g$ for all $g\ne 5$. As an application we show that every 6-edge-connected planar (multi)graph contains two edge-disjoint $\frac{18}{19}$-thin spanning trees.

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arXiv2016

Reconstruction of infinite matroids from their 3-connected minors

We show that any infinite matroid can be reconstructed from the torsos of a tree-decomposition over its 2-separations, together with local information at the ends of the tree. We show that if the matroid is tame then this local information is simply a choice of whether circuits are permitted to use that end. The same is true if each torso is planar, with all gluing elements on a common face.

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arXiv2016

Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a graph on $n$ vertices embedded on a surface $Σ$ of genus $g$, then there exist constants $ε,c_g > 0$ such that if $G$ has an $L$-coloring, then $G$ has at least $c_g2^{εn}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. More generally, they proved that there exist constants $ε,α>0$ such that if $G$ is a graph on $n$ vertices embedded in a surface $Σ$ of fixed genus $g$, $H$ is a proper subgraph of $G$, and $ϕ$ is an $L$-coloring of $H$ that extends to an $L$-coloring of $G$, then $ϕ$ extends to at least $2^{ε(n - α(g + |V(H)|))}$ distinct $L$-colorings of $G$ if $L$ is a 5-list-assignment or if $L$ is a 3-list-assignment and $G$ has girth at least five. We prove the same result if $G$ is triangle-free and $L$ is a 4-list-assignment of $G$, where $ε=\frac{1}{8}$, and $α= 130$.

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arXiv2016

5-list-coloring planar graphs with distant precolored vertices

We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.

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arXiv2015

Modified Linear Programming and Class 0 Bounds for Graph Pebbling

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.

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arXiv2015

The structure of 2-separations of infinite matroids

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.

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arXiv2015

Five-list-coloring graphs on surfaces II. A linear bound for critical graphs in a disk

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $ϕ$ of $J$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $J$. We prove a conjecture of Dvorak et al. that if $H$ is a minimal subgraph of $G$ such that $C$ is a subgraph of $H$ and every $L$-coloring of $C$ that extends to an $L$-coloring of $H$ also extends to an $L$-coloring of $G$, then $|V(H)|\le 19|V(C)|$. This is a lemma that plays an important role in subsequent papers, because it motivates the study of graphs embedded in surfaces that satisfy an isoperimetric inequality suggested by this result. Such study turned out to be quite profitable for the subject of list coloring graphs on surfaces.

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arXiv2014

Density of 5/2-critical graphs

A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C_5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n>=4 vertices has at least (5n-2)/4 edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.

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arXiv2014

On the Minimum Edge-Density of 4-Critical Graphs of Girth Five

We prove that if G is a 4-critical graph of girth at least five then |E(G)|>=(5|V(G)|+2)/3. As a corollary, graphs of girth at least five embeddable in the Klein bottle or torus are 3-colorable. These are results of Thomas and Walls, and Thomassen respectively. The proof uses the new potential technique developed by Kostochka and Yancey who proved that 4-critical graphs satisfy: |E(G)|>=(5|V(G)|-2)/3.

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arXiv2014

Characterizing 4-Critical Graphs of Ore-Degree at most Seven

Dirac introduced the notion of a k-critical graph, a graph that is not (k-1)-colorable but whose every proper subgraph is (k-1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a subgraph isomorphic to K_{k+1} (or an odd cycle for k=2). Equivalently, for all k>=4, the only k-critical graph of maximum degree k-1 is K_k. A natural generalization of Brook's theorem is to consider the Ore-degree of a graph, which is the maximum of d(u)+d(v) over all edges uv. Kierstead and Kostochka proved that for all k>=6 the only k-critical graph with Ore-degree at most 2k-1 is K_k. Kostochka, Rabern and Steibitz proved that the only 5-critical graphs with Ore-degree at most 9 are K_5 and a graph they called O_5. A different generalization of Brook's theorem, motivated by Hajos' construction, is Gallai's conjectured bound on the minimum density of a k-critical graph. Recently, Kostochka and Yancey proved Gallai's conjecture. Their proof for k>=5 implies the above results on Ore-degree. However, the case for k=4 remains open, which is the subject of this paper. Kostochka and Yancey's short but beautiful proof for the case k=4 says that if $G$ is a $4$-critical graph, then |E(G)|>= (5|V(G)|-2)/3. We prove the following bound which is better when there exists a large independent set of degree three vertices: if G is a 4-critical graph G, then |E(G)|>= 1.6 |V(G)| + .2 alpha(D_3(G)) - .6, where D_3(G) is the graph induced by the degree three vertices of G. As a corollary, we characterize the 4-critical graphs with Ore-degree at most seven as precisely the graphs of Ore-degree seven in the family of graphs obtained from K_4 and Ore compositions.

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