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Bojan Mohar

Publications and source records attributed to Bojan Mohar.

At least 55 records · Page 3Linked to original sources
arXiv2024

Cops and Robber on Hyperbolic Manifolds

The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus $g$ have largest cop number. It is known that the cop number of genus $g$ graphs is $O(g)$ and that there are examples whose cop number is $\tildeΩ(\sqrt{g}\,)$. The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result about the game on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. We also discuss upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds.

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arXiv2023

Expected number of faces in a random embedding of any graph is at most linear

A random 2-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log(n)$. While there are many families of graphs whose expected number of faces is $Θ(n)$, none are known where the expected number would be super-linear. This lead to the conjecture that there is a linear upper bound. In this note we confirm the conjecture by proving that for any $n$-vertex multigraph, the expected number of faces in a random 2-cell embedding is at most $n(1+H_m)$, where $m$ is the maximum edge-multiplicity and $H_m$ denotes the $m$th harmonic number. This bound is best possible up to a constant factor.

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arXiv2023

Subdivision and Graph Eigenvalues

This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the $k$-th largest and $k$-th smallest eigenvalues for any fixed $k$. It is expected that after subdivision, most eigenvalues of the new graph will lie in the interval $[-2,2]$. We examine the eigenvalues of the new graph outside this interval, and we prove several results that might be of independent interest.

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arXiv2022

Treewidth, Circle Graphs and Circular Drawings

A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs, and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. Using the same tools, we also study the treewidth of graphs $G$ that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, we show that if the crossing graph is $K_t$-minor-free, then $G$ has treewidth at most $12t-23$ and has no $K_{2,4t}$-topological minor. On the other hand, we show that there are graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are $2$-degenerate.

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arXiv2022

Simple eigenvalues of cubic vertex-transitive graphs

If $v$ is an eigenvector for eigenvalue $λ$ of a graph $X$ and $α$ is an automorphism of $X$, then $α(v)$ is also an eigenvector for $λ$. Thus it is rather exceptional for an eigenvalue of a vertex-transitive graph to be simple. We study cubic vertex-transitive graphs with a non-trivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps and Chebyshev polynomials.

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arXiv2022

Proper orientations and proper chromatic number

The proper chromatic number $\Vecχ(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper chromatic number are resolved. First it is shown, that $\Vecχ(G)$ of any planar graph $G$ is bounded (in fact, it is at most 14). Secondly, it is shown that for every graph, $\Vecχ(G)$ is at most $O(\frac{r\log r}{\log\log r})+\tfrac{1}{2}\MAD(G)$, where $r=χ(G)$ is the usual chromatic number of the graph, and $\MAD(G)$ is the maximum average degree taken over all subgraphs of $G$. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations.

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arXiv2022

On density of $Z_3$-flow-critical graphs

For an abelian group $Γ$, a graph $G$ is said to be $Γ$-flow-critical if $G$ does not admit a nowhere-zero $Γ$-flow, but for each edge $e\in E(G)$, the contraction $G/e$ has a nowhere-zero $Γ$-flow. A bound on the density of $Z_3$-flow-critical graphs drawn on a fixed surface is obtained, generalizing the planar case of the bound on the density of 4-critical graphs by Kostochka and Yancey.

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arXiv2022

Graphs of Linear Growth have Bounded Treewidth

A graph class $\mathcal{G}$ has linear growth if, for each graph $G \in \mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.

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arXiv2022

Inducibility and universality for trees

We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive $\varepsilon_1$ and $\varepsilon_2$ such that every tree that is neither a path nor a star has inducibility at most $1-\varepsilon_1$, where the inducibility of a tree $T$ is defined as the maximum limit density of $T$, and that there are infinitely many trees with inducibility at least $\varepsilon_2$. Finally, we construct a universal sequence of trees; that is, a sequence in which the limit density of any tree is positive.

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arXiv2022

The game of Cops and Robber on geodesic spaces

The game of Cops and Robber is traditionally played on a finite graph. The purpose of this paper is to introduce and analyse the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer $c$ such that $c$ cops can win the game against one robber, and $c$ only depends on the genus $g$ of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=Ω(\sqrt{g})$ when the genus $g$ is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.

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arXiv2021

Min-max theorem for the game of Cops and Robber on geodesic spaces

The game of Cops and Robber is traditionally played on a finite graph. The purpose of this note is to introduce and analyze the game that is played on an arbitrary geodesic space. The game is defined in such a way that it preserves the beauty and power of discrete games played on graphs and also keeps the specialties of the pursuit-evasion games played on polyhedral complexes. It is shown that the game can be approximated by finite games of discrete type and as a consequence a min-max theorem is obtained.

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arXiv2021

Random 2-cell embeddings of multistars

Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with $n$ nonleaf edges lies in an interval of length $2/(n + 1)$ centered at the expected number of faces of an $n$-edge dipole. This allows us to derive bounds on $\mathbb{E}[F_G]$ for any given graph $G$ in terms of vertex degrees. We conjecture that $\mathbb{E}[F_G ] \le O(n)$ for any simple $n$-vertex graph $G$.

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arXiv2021

Multicoloured Ramsey numbers of the path of length four

Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a monochromatic copy of $P_t$. In this note, we determine $R_r(P_5)$, which is approximately $3r$.

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arXiv2021

A rainbow connectivity threshold for random graph families

Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in $\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each sampled randomly from $G(n,p)$, with $n = |X|$ and $p = \frac{c \log n}{sn}$, where $c > 1$ is a constant. We show that when $s$ is sufficiently large, $\mathcal G$ is a.a.s. rainbow connected, and when $s$ is sufficiently small, $\mathcal G$ is a.a.s. not rainbow connected. We also calculate a threshold of $s$ for the rainbow connectivity of $\mathcal G$, and we show that this threshold is concentrated on at most three values, which are larger than the diameter of the union of $\mathcal G$ by about $\frac{\log n}{(\log \log n)^2}$. The same results also hold in a more traditional random rainbow setting, where we take a random graph $G\in G(n,p)$ with $p=\frac{c \log n}{n}$ ($c>1$) and color each edge of $G$ with a color chosen uniformly at random from the set $[s]$ of $s$ colors.

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arXiv2021

Bounded maximum degree conjecture holds precisely for $c$-crossing-critical graphs with $c \leq 12$

We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For every fixed pair of integers with $c\ge 13$ and $d\ge 1$, we give first explicit constructions of $c$-crossing-critical graphs containing a vertex of degree greater than $d$. We also show that such unbounded degree constructions do not exist for $c\le 12$, precisely, that there exists a constant $D$ such that every $c$-crossing-critical graph with $c\le 12$ has maximum degree at most $D$. Hence, the bounded maximum degree conjecture of $c$-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values $c\le 12.$

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arXiv2021

Automorphism groups of maps in linear time

By a map we mean a $2$-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. Automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.

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arXiv2020

Meyniel's conjecture on graphs of bounded degree

The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set $\{c(G) \mid G \text{ cubic}\}$ is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1/2-o(1)}$ where $n=|V(G)|$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.

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