arXiv subjects

Bojan Mohar

Publications and source records attributed to Bojan Mohar.

At least 145 records · Page 8Linked to original sources

arXiv2011

Two results on the digraph chromatic number

It is known (Bollobás (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $Δ$ and of arbitrarily large girth whose chromatic number is at least $c Δ/ \log Δ$. We show an analogous result for digraphs where the chromatic number of a digraph $D$ is defined as the minimum integer $k$ so that $V(D)$ can be partitioned into $k$ acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdos (1962), that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

math.CO↗

arXiv2011

Planar graphs have exponentially many 3-arboricities

It is well-known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any (projective) planar graph. The same result holds in the setting of 3-list-colorings.

math.CO↗

arXiv2011

Uniquely D-colourable digraphs with large girth

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

math.CO↗

arXiv2011

Immersing complete digraphs

We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum degree at least t-1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high in- and outdegree which do not contain an immersion of the complete digraph on 3 vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construction of such graphs by Thomassen.

math.CO↗

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.

math.CO↗

arXiv2010

Expected Crossing Numbers

The expected value for the weighted crossing number of a randomly weighted graph is studied. A variation of the Crossing Lemma for expectations is proved. We focus on the case where the edge-weights are independent random variables that are uniformly distributed on [0,1].

math.CO↗

arXiv2010

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph was introduced by Erdős et al. [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs's result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.

math.CO↗

arXiv2010

Unexpected behaviour of crossing sequences

The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.

math.CO↗

arXiv2010

On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice

We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at zero temperature. We prove that this algorithm is not ergodic for symmetric subsets of the kagome lattice with fully periodic boundary conditions: given an initial configuration, not all configurations are accessible via Monte Carlo steps. The same conclusion holds for single-site dynamics.

cond-mat.stat-mech↗

arXiv2010

Kempe equivalence of edge-colourings in subcubic and subquartic graphs

It is proved that all 4-edge-colourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Delta=3 is a threshold for Kempe equivalence of (Delta+1)-edge-colourings, as such an equivalence does not hold in general when Delta=4. One extra colour allows a similar result in this latter case however, namely, when Delta<=4 it is shown that all (Delta+2)-edge-colourings are Kempe equivalent.

math.CO↗

arXiv2010

An Eberhard-like theorem for pentagons and heptagons

Eberhard proved that for every sequence $(p_k), 3\le k\le r, k\ne 5,7$ of non-negative integers satisfying Euler's formula $\sum_{k\ge3} (6-k) p_k = 12$, there are infinitely many values $p_6$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge3$, where $p_k=0$ if $k>r$. In this paper we prove a similar statement when non-negative integers $p_k$ are given for $3\le k\le r$, except for $k=5$ and $k=7$. We prove that there are infinitely many values $p_5,p_7$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge3$. %, where $p_k=0$ if $k>r$. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind.

math.CO↗

arXiv2010

A strengthening and a multipartite generalization of the Alon-Boppana-Serre Theorem

The Alon-Boppana theorem confirms that for every $ε>0$ and every integer $d\ge3$, there are only finitely many $d$-regular graphs whose second largest eigenvalue is at most $2\sqrt{d-1}-ε$. Serre gave a strengthening showing that a positive proportion of eigenvalues of any $d$-regular graph must be bigger than $2\sqrt{d-1}-ε$. We provide a multipartite version of this result. Our proofs are elementary and work also in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree $d$ and bounded maximum degree. The two-partite result shows that for every $ε>0$ and any positive integers $d_1,d_2,d$, every $n$-vertex graph of maximum degree at most $d$, whose vertex set is the union of (not necessarily disjoint) subsets $V_1,V_2$, such that every vertex in $V_i$ has at least $d_i$ neighbors in $V_{3-i}$ for $i=1,2$, has $Ω_ε(n)$ eigenvalues that are larger than $\sqrt{d_1-1}+\sqrt{d_2-1}-ε$. Finally, we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound $2\sqrt{d-1}-ε$ can be replaced by $2\sqrt{d-1} + δ$ for some $δ>0$ if graphs have bounded "global girth". On the other side of the spectrum, if the odd girth is large, then we get an Alon-Boppana-Serre type theorem for the negative eigenvalues as well.

math.CO↗

arXiv2010

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}γ^n n!$ where $c^{(g)}>0$, and $γ\approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability.

math.CO↗

arXiv2009

Strong embeddings of minimum genus

A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.

math.CO↗

arXiv2009

The excluded minor structure theorem with planarly embedded wall

A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly embedded graph, many rows of which intersect the embedded subgraph $G_0$ of the near-embedding, contains a large subwall that is planarly embedded within $G_0$. This result provides some hidden details needed for a strong version of the Robertson and Seymour's excluded minor theorem as presented in [K. Kawarabayashi, B. Mohar, Some recent progress and applications in graph minor theory, Graphs Combin. 23 (2007) 1-46].

math.CO↗

arXiv2009

Chromatic number and complete graph substructures for degree sequences

Given a graphic degree sequence $D$, let $χ(D)$ (respectively $ω(D)$, $h(D)$, and $H(D)$) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is $D$. It is proved that $χ(D)\le h(D)$. Moreover, it is shown that a subdivision of a clique of order $χ(D)$ exists where each edge is subdivided at most once and the set of all subdivided edges forms a collection of disjoint stars. This bound is an analogue of the Hajos Conjecture for degree sequences and, in particular, settles a conjecture of Neil Robertson that degree sequences satisfy the bound $χ(D)\le H(D)$ (which is related to the Hadwiger Conjecture). It is also proved that $χ(D)\le {6/5}ω(D)+{3/5}$ and that $χ(D) \le {4/5}ω(D) + {1/5}Δ(D) + 1$, where $Δ(D)$ denotes the maximum degree in $D$. The latter inequality is a strengthened version of a conjecture of Bruce Reed. All derived inequalities are best possible.

math.CO↗

arXiv2009

Crossing-critical graphs with large maximum degree

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In this note we disprove these conjectures for every $k\ge 171$, by providing examples of $k$-crossing-critical graphs with arbitrarily large maximum degree.

math.CO↗

arXiv2009

Spectral radius of finite and infinite planar graphs and of graphs of bounded genus

It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius $ρ(G)$ of a planar graph $G$ of maximum vertex degree $D\ge 4$ satisfies $\sqrt{D}\le ρ(G)\le \sqrt{8D-16}+7.75$. This result is best possible up to the additive constant--we construct an (infinite) planar graph of maximum degree $D$, whose spectral radius is $\sqrt{8D-16}$. This generalizes and improves several previous results and solves an open problem proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus. For every $k$, these bounds can be improved by excluding $K_{2,k}$ as a subgraph. In particular, the upper bound is strengthened for 5-connected graphs. All our results hold for finite as well as for infinite graphs. At the end we enhance the graph decomposition method introduced in the first part of the paper and apply it to tessellations of the hyperbolic plane. We derive bounds on the spectral radius that are close to the true value, and even in the simplest case of regular tessellations of type $\{p,q\}$ we derive an essential improvement over known results, obtaining exact estimates in the first order term and non-trivial estimates for the second order asymptotics.

math.CO↗