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Zdeněk Dvořák

Publications and source records attributed to Zdeněk Dvořák.

87 records · Page 5Linked to original sources
arXiv2017

Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains

Aksenov proved that in a planar graph G with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact characterization of planar graphs with two triangles in that some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in followup papers to give detailed information about the structure of 4-critical triangle-free graphs embedded in a fixed surface.

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arXiv2017

Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon) for some fixed epsilon>0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G' such that alpha(G')-|G'|/3=alpha(G)-|G|/3 and |G'|<=(alpha(G)-|G|/3)/epsilon. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane triangle-free graphs whose independence number is close to n/3.

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arXiv2016

Triangle-free planar graphs with the smallest independence number

Steinberg and Tovey proved that every n-vertex planar triangle-free graph has an independent set of size at least (n+1)/3, and described an infinite class of tight examples. We show that all n-vertex planar triangle-free graphs except for this one infinite class have independent sets of size at least (n+2)/3.

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arXiv2015

Complete graph immersions and minimum degree

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.

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arXiv2015

Fine structure of 4-critical triangle-free graphs II. Planar triangle-free graphs with two precolored 4-cycles

We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ such that if $H$ is a planar triangle-free graph and $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a 3-coloring of $H$. This gives a positive answer to a conjecture of Dvořák, Král' and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree.

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arXiv2014

4-critical graphs on surfaces without contractible (<=4)-cycles

We show that if G is a 4-critical graph embedded in a fixed surface $Σ$ so that every contractible cycle has length at least 5, then G can be expressed as $G=G'\cup G_1\cup G_2\cup ... \cup G_k$, where $|V(G')|$ and $k$ are bounded by a constant (depending linearly on the genus of $Σ$) and $G_1\ldots,G_k$ are graphs (of unbounded size) whose structure we describe exactly. The proof is computer-assisted - we use computer to enumerate all plane 4-critical graphs of girth 5 with a precolored cycle of length at most 16, that are used in the basic case of the inductive proof of the statement.

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arXiv2013

Immersion in four-edge-connected graphs

Fix g>1. Every graph of large enough tree-width contains a g x g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g x g grid as an immersion (and hence contains any fixed graph with maximum degree at most four as an immersion). This result has a number of applications.

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arXiv2013

Planar 4-critical graphs with four triangles

By the Grunbaum-Aksenov Theorem (extending Grotzsch's Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdos from 1990.

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arXiv2011

Star chromatic index

The star chromatic index $χ_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree $Δ=Δ(G)$. Our best lower bound on $χ_s'$ in terms of $Δ$ is $2Δ(1+o(1))$ valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

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arXiv2011

Graphs with two crossings are 5-choosable

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.

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arXiv2011

Minimum degree condition forcing complete graph immersion

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and $f(v)$. The immersion is strong if the paths $P_{uv}$ are internally disjoint from $f(V(H))$. It is proved that for every positive integer $t$, every simple graph of minimum degree at least $200t$ contains a strong immersion of the complete graph $K_t$. For dense graphs one can say even more. If the graph has order $n$ and has $2cn^2$ edges, then there is a strong immersion of the complete graph on at least $c^2 n$ vertices in $G$ in which each path $P_{uv}$ is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree $d$ has a clique minor of order at least $cd^{3/2}$, where $c>0$ is an absolute constant. For small values of $t$, $1\le t\le 7$, every simple graph of minimum degree at least $t-1$ contains an immersion of $K_t$ (Lescure and Meyniel, DeVos et al.). We provide a general class of examples showing that this does not hold when $t$ is large.

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