arXiv subjects

Jochen Harant

Publications and source records attributed to Jochen Harant.

10 recordsLinked to original sources

arXiv2025

Lower bounds on the independence number of a graph in terms of degrees

Given an integer $\Delta \ge 3$, let ${\cal G}_{\Delta }$ be the set of connected graphs $G\neq K_{\Delta +1}$ with maximum degree $\Delta $ and, for $i=1,\cdots, \Delta $, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. \\ We prove that $\sum\limits_{i=1}^\Delta c_i|V_i(G)|$ is a lower bound on the independence number $\alpha(G)$ of $G\in {\cal G}_\Delta$, where $c_\Delta=\frac{1}{\Delta}$ and $ic_{i}=1-c_{i+1}$ for $i=1,\cdots,\Delta-1$. Moreover, if $\varepsilon >0$ and $j\in \{1,\cdots, \Delta\}$, then the inequality $\alpha(G)\ge \varepsilon|V_j(G)|+\sum\limits_{i=1}^\Delta c_i|V_i(G)|$ does not hold for infinitely many graphs $G\in {\cal G}_\Delta$. We also show that an independent set $I\subset V(G)$ of $G\in {\cal G}_\Delta$ such that $|I|\ge \sum\limits_{i=1}^\Delta c_i|V_i(G)|$ can be found in polynomial time.

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arXiv2025

On the independence number in subcubic graphs

For a connected subcubic graph $G\neq K_1$ let $V_i(G) = \{v \in V(G) ~|~ d_G(v)=i\}$ for $1 \leq i \leq 3.$ Given $c_1, c_2, c_ 3 \in \mathbb{R}^+$ and $ d \in \mathbb{R}$, we show several results of type $\alpha(G) \geq c_1|V_1(G)| + c_2|V_2(G)| + c_3|V_3(G)| - d.$ We also derive classes of graphs $G$ showing sharpness of these lower bounds on the independence number $\alpha(G)$ of $G$.

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arXiv2022

Minimum Spanning Trees with Bounded Degrees of Vertices in a Specified Stable Set

Given a graph $G$ and sets $\{\alpha_v~|~v \in V(G)\}$ and $\{\beta_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in V(G)$ is $NP$-complete. In this article, we relax the problem by demanding that the degree restrictions apply to vertices $v\in U$ only, where $U$ is a stable set of $G$. In this case, the problem becomes tractable. A. Frank presented a result characterizing the positive instances of that relaxed problem. Using matroid intersection developed by J. Edmonds, we give a new and short proof of Frank's result and show that if $U$ is stable and the edges of $G$ are weighted by arbitrary real numbers, then even a minimum-cost tree $T$ with $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in U$ can be found in polynomial time if such a tree exists.

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arXiv2021

Circumference of essentially 4-connected planar triangulations

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{3}(n+4)$; moreover, this bound is sharp.

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arXiv2020

New bounds on domination and independence in graphs

We propose new bounds on the domination number and on the independence number of a graph and show that our bounds compare favorably to recent ones. Our bounds are obtained by using the Bhatia-Davis inequality linking the variance, the expected value, the minimum, and the maximum of a random variable with bounded distribution.

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arXiv2020

Rooted Minors and Locally Spanning Subgraphs

Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider "local" versions of such statements. In 1966, for instance, D. W. Barnette proved that a $3$-connected planar graph contains a spanning tree of maximum degree at most $3$. A local translation of this statement is that if $G$ is a planar graph, $X$ is a subset of specified vertices of $G$ such that $X$ cannot be separated in $G$ by removing $2$ or fewer vertices of $G$, then $G$ has a tree of maximum degree at most $3$ containing all vertices of $X$. Our results constitute a general machinery for strengthening statements about $k$-connected graphs (for $1 \leq k \leq 4$) to locally spanning versions, i.e. subgraphs containing a set $X\subseteq V(G)$ of a (not necessarily planar) graph $G$ in which only $X$ has high connectedness. Given a graph $G$ and $X\subseteq V(G)$, we say $M$ is a minor of $G$ rooted at $X$, if $M$ is a minor of $G$ such that each bag of $M$ contains at most one vertex of $X$ and $X$ is a subset of the union of all bags. We show that $G$ has a highly connected minor rooted at $X$ if $X\subseteq V(G)$ cannot be separated in $G$ by removing a few vertices of $G$. Combining these investigations and the theory of Tutte paths in the planar case yields to locally spanning versions of six well-known results about degree-bounded trees, hamiltonian paths and cycles, and $2$-connected subgraphs of graphs.

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arXiv2019

Long Cycles and Spanning Subgraphs of Locally Maximal 1-planar Graphs

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal $1$-planar. For a $3$-connected locally maximal $1$-planar graph $G$, we show the existence of a spanning $3$-connected planar subgraph and prove that $G$ is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that $G$ is traceable if $G$ has at most four $3$-vertex-cuts. Moreover, infinitely many non-traceable $5$-connected $1$-planar graphs are presented.

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arXiv2018

On the Circumference of Essentially 4-connected Planar Graphs

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.

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arXiv2017

Longer Cycles in Essentially 4-Connected Planar Graphs

A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm circ}\nolimits(G)$ of a longest cycle of any essentially 4-connected planar graph $G$ on $n$ vertices is at least $\frac{2n+4}{5}$ and Fabrici, Harant and Jendrol' improved this result to $\mathop{\rm circ}\nolimits(G)\geq \frac{1}{2}(n+4)$. In the present paper, we prove that an essentially 4-connected planar graph on $n$ vertices contains a cycle of length at least $\frac{3}{5}(n+2)$ and that such a cycle can be found in time $O(n^2)$.

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arXiv2017

On Selkow's Bound on the Independence Number of Graphs

For a graph $G$ with vertex set $V(G)$ and independence number $\alpha(G)$, S. M. Selkow (Discrete Mathematics, 132(1994)363--365) established the famous lower bound $\sum\limits_{v\in V(G)}\frac{1}{d(v)+1}(1+\max\{\frac{d(v)}{d(v)+1}-\sum\limits_{u\in N(v)}\frac{1}{d(u)+1},0 \})$ on $\alpha(G)$, where $N(v)$ and $d(v)=|N(v)|$ denote the neighborhood and the degree of a vertex $v\in V(G)$, respectively. However, Selkow's original proof of this result is incorrect. We give a new probabilistic proof of Selkow's bound here.

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