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Raphael Steiner

Publications and source records attributed to Raphael Steiner.

At least 73 records · Page 4Linked to original sources
arXiv2022

Strengthening Hadwiger's conjecture for $4$- and $5$-chromatic graphs

Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a $5$-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.

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arXiv2022

Coloring Drawings of Graphs

We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $Γ(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring of $Γ(G)$ to be a coloring of the maximal connected regions of the drawing, the cells, with $k$ colors such that adjacent cells have different colors. By the $4$-color theorem, every drawing of a bridgeless graph has a cell $4$-coloring. A drawing of a graph is cell $2$-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a cell $3$-colorable drawing. This leads to the natural question which abstract graphs have the property that each of their drawings has a cell $3$-coloring. We say that such a graph is universally cell $3$-colorable. We show that every $4$-edge-connected graph and every graph admitting a nowhere-zero $3$-flow is universally cell $3$-colorable. We also discuss circumstances under which universal cell $3$-colorability guarantees the existence of a nowhere-zero $3$-flow. On the negative side, we present an infinite family of universally cell $3$-colorable graphs without a nowhere-zero $3$-flow. On the positive side, we formulate a conjecture which has a surprising relation to a famous open problem by Tutte known as the $3$-flow-conjecture. We prove our conjecture for subcubic and for $K_{3,3}$-minor-free graphs.

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arXiv2022

Matching Theory and Barnette's Conjecture

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity. As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985]. These allow us to check whether a graph we generated is a brace.

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arXiv2022

Subdivisions with congruence constraints in digraphs of large chromatic number

We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e \in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.

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arXiv2022

Exact Matching in Graphs of Bounded Independence Number

In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer $k$. The task is then to decide whether the given graph contains a perfect matching exactly $k$ of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems. When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be $\textbf{NP}$-complete. Later however, Mulmuley et al.~presented a randomized polynomial time algorithm for EM, which puts EM in $\textbf{RP}$. Given that to decide whether or not $\textbf{RP}=\textbf{P}$ represents a big open challenge in complexity theory, this makes it unlikely for EM to be $\textbf{NP}$-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in $\textbf{RP}$ which are not known to be contained in $\textbf{P}$, making it an interesting instance for testing the hypothesis $\textbf{RP}=\textbf{P}$. Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we finally push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs.

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arXiv2022

Colorings of oriented planar graphs avoiding a monochromatic subgraph

For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, and that for every fixed digraph $F$ with $Δ(F) \ge 3$, every oriented planar graph $D$ admits an $F$-free $3$-coloring. We show in contrast, that - if $F$ is an orientation of a path of length at least $2$, then it is NP-hard to decide whether an acyclic and planar input digraph $D$ admits an $F$-free $2$-coloring. - if $F$ is an orientation of a path of length at least $1$, then it is NP-hard to decide whether an acyclic and planar input digraph $D$ admits an $F$-free $3$-coloring.

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arXiv2022

Coloring circle arrangements: New $4$-chromatic planar graphs

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most $3$. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps $\triangle$-saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This "corona" construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture. We also investigate fractional colorings. It is shown that the arrangement graph of every arrangement $\mathcal{A}$ of pairwise intersecting pseudocircles is "close" to being $3$-colorable. More precisely, the fractional chromatic number $χ_f(\mathcal{A})$ of the arrangement graph is bounded from above by $χ_f(\mathcal{A}) \le 3+O(\frac{1}{n})$, where $n$ is the number of pseudocircles of $\mathcal{A}$. Furthermore, we construct an infinite family of $4$-edge-critical $4$-regular planar graphs which are fractionally $3$-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019).

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arXiv2022

Cycle lengths modulo $k$ in expanders

Given a constant $α>0$, an $n$-vertex graph is called an $α$-expander if every set $X$ of at most $n/2$ vertices in $G$ has an external neighborhood of size at least $α|X|$. Addressing a question posed by Friedman and Krivelevich in [Combinatorica, 41(1), (2021), pp. 53--74], we prove the following result: Let $k>1$ be an integer with smallest prime divisor $p$. Then for $α>\frac{1}{p-1}$ every sufficiently large $α$-expanding graph contains cycles of length congruent to any given residue modulo $k$. This result is almost best possible, in the following sense: There exists an absolute constant $c>0$ such that for every integer $k$ with smallest prime divisor $p$ and for every positive $α<\frac{c}{p-1}$, there exist arbitrarily large $α$-expanding graphs with no cycles of length $r$ modulo $k$, for some $r \in \{0,\ldots,k-1\}$.

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arXiv2022

Improved bound for improper colorings of graphs with no odd clique minor

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-coloring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colors is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008), Kang and Oum (2019), Liu and Wood (2021), and strengthens a result by van den Heuvel and Wood (2018), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$-minor free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was non-explicit. Our short proof combines the method by van den Heuvel and Wood for $K_t$-minor free graphs with some additional ideas, which make the extension to odd $K_t$-minor free graphs possible.

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arXiv2022

Disproof of a Conjecture by Woodall

In 2001, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon>0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such that for all integers $s,t $ with $N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\varepsilon)(2s+t)$.

math.CO↗
arXiv2021

Hat guessing numbers of strongly degenerate graphs

Assume $n$ players are placed on the $n$ vertices of a graph $G$. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a colour out of $q$ available colours. The players can see the hat of each of their neighbours in $G$, but not their own hat. Using a prediscussed guessing strategy, the players then simultaneously guess the colour of their hat. The players win if at least one of them guesses correctly, else the adversary wins. The largest integer $q$ such that there is a winning strategy for the players is denoted by $\text{HG}(G)$, and this is called the hat guessing number of $G$. Although this game has received a lot of attention in the recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterisation of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from $2^{125000}$ to $40$, and further derive upper bounds on the hat guessing number for any class of $K_{2,s}$-free graphs with bounded expansion, such as the class of $C_4$-free planar graphs, more generally $K_{2,s}$-free graphs with bounded Hadwiger number or without a $K_t$-subdivision, and for Erdős-Rényi random graphs with constant average degree.

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arXiv2021

Edge Partitions of Complete Geometric Graphs (Part 2)

Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

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arXiv2021

Tight bounds for divisible subdivisions

Alon and Krivelevich proved that for every $n$-vertex subcubic graph $H$ and every integer $q \ge 2$ there exists a (smallest) integer $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which the length of every subdivision-path is divisible by $q$. Improving their superexponential bound, we show that $f(H,q) \le \frac{21}{2}qn+8n+14q$, which is optimal up to a constant multiplicative factor.

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arXiv2021

Improved lower bound for the list chromatic number of graphs with no $K_t$ minor

Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$. Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$. Our lower-bound examples are obtained via the probabilistic method.

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arXiv2021

Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant

Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd $K_t$-minor is $(t-1)$-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form $Ct$ for some constant $C>0$ exists. We show that if every graph without a $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, the recent $O(t\log\log t)$-upper bound of Delcourt and Postle for the chromatic number of $K_t$-minor free graphs directly carries over to the chromatic number of odd $K_t$-minor-free graphs. This (slightly) improves a previous bound of $O(t(\log \log t)^2)$ for this problem by Delcourt and Postle.

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arXiv2021

Zero sum cycles in complete digraphs

Given a non-trivial finite Abelian group $(A,+)$, let $n(A) \ge 2$ be the smallest integer such that for every labelling of the arcs of the bidirected complete graph of order $n(A)$ with elements from $A$ there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining $n(\mathbb{Z}_q)$ for integers $q \ge 2$ was recently considered by Alon and Krivelevich, who proved that $n(\mathbb{Z}_q)=O(q \log q)$. Here we improve their bound and show that $n(\mathbb{Z}_q)$ grows linearly. More generally we prove that for every finite Abelian group $A$ we have $n(A) \le 8|A|$, while if $|A|$ is prime then $n(A) \le \frac{3}{2}|A|$. As a corollary we also obtain that every $K_{16q}$-minor contains a cycle of length divisible by $q$ for every integer $q \ge 2$, which improves a result by Alon and Krivelevich.

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arXiv2021

On coloring digraphs with forbidden induced subgraphs

We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every oriented graph in which the out-neighborhood of every vertex induces a transitive tournament can be partitioned into two acyclic induced subdigraphs. We prove multiple extensions of this result to larger classes of digraphs defined by a finite list of forbidden induced subdigraphs. We thereby resolve several special cases of an extension of the famous Gyárfás-Sumner conjecture to directed graphs by Aboulker et al.

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