arXiv subjects
Publications and source records attributed to Zdenek Dvorak.
A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(sqrt{n}) edges. This bound is asymptotically optimal.
Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct 3-colorings.
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.
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.
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.