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Dieter Rautenbach

Publications and source records attributed to Dieter Rautenbach.

At least 37 records · Page 2Linked to original sources
arXiv2023

Sparse vertex cutsets and the maximum degree

We show that every graph $G$ of maximum degree $Δ$ and sufficiently large order has a vertex cutset $S$ of order at most $Δ$ that induces a subgraph $G[S]$ of maximum degree at most $Δ-3$. For $Δ\in \{ 4,5\}$, we refine this result by considering also the average degree of $G[S]$. If $G$ has no $K_{r,r}$ subgraph, then we show the existence of a vertex cutset that induces a subgraph of maximum degree at most $\left(1-\frac{1}{r\choose 2}\right)Δ+O(1)$.

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arXiv2023

Irregularity of Graphs respecting Degree Bounds

Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $Δ$, and $d=\left\lfloor \frac{Δm}{Δn-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}Δ\left(Δ^2-(2d+1)Δ-d^2-d\right)m.$$

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arXiv2023

Vertex degrees close to the average degree

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular, for $d_+\in \left(\sqrt{dn},n-1\right]$, we show the existence of a vertex in $G$ of degree between $d_+-\left(\frac{(d_+-d)n}{n-d_++\sqrt{d_+^2-dn}}\right)$ and $d_+$.

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arXiv2022

Bounding the Mostar index

Došlić et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. They conjectured that $Mo(G)\leq 0.\overline{148}n^3$ for every graph $G$ of order $n$. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter $Mo^\star(G)=\sum\limits_{uv\in E(G)}\big(n-\min\{ d_G(u),d_G(v)\}\big)$. For a graph $G$ of order $n$, they show that $Mo^\star(G)\leq \frac{5}{24}(1+o(1))n^3$. We improve this bound to $Mo^\star(G)\leq \left(\frac{2}{\sqrt{3}}-1\right)n^3$, which is best possible up to terms of lower order. Furthermore, we show that $Mo^\star(G)\leq \left(2\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)-2\left(\fracΔ{n}\right)\sqrt{\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)}\right)n^3$ provided that $G$ has maximum degree $Δ$.

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arXiv2022

Maximizing the Mostar index for bipartite graphs and split graphs

Došlić et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.

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arXiv2022

Majority Edge-Colorings of Graphs

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.

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arXiv2022

Relating the independence number and the dissociation number

The independence number $α(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements of the obvious inequality $2α(G)\geq {\rm diss}(G)$. For connected cubic graphs $G$ distinct from $K_4$, we show $5α(G)\geq 3{\rm diss}(G)$, and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.

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arXiv2022

Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree

We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq τ(u)\}$ for every positive integer $t$, and $τ:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,τ)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $τ$ satisfying $τ(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.

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arXiv2022

Efficiently recognizing graphs with equal independence and annihilation numbers

The annihilation number $a(G)$ of a graph $G$ is an efficiently computable upper bound on the independence number $α(G)$ of $G$. Recently, Hiller observed that a characterization of the graphs $G$ with $α(G)=a(G)$ due to Larson and Pepper is false. Since the known efficient algorithm recognizing these graphs was based on this characterization, the complexity of recognizing graphs $G$ with $α(G)=a(G)$ was once again open. We show that these graphs can indeed be recognized efficiently. More generally, we show that recognizing graphs $G$ with $α(G)\geq a(G)-\ell$ is fixed parameter tractable using $\ell$ as parameter.

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arXiv2022

A bound on the dissociation number

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph $G$ with $n$ vertices, $m$ edges, $k$ components, and $c_1$ induced cycles of length $1$ modulo $3$, we show ${\rm diss}(G)\geq n-\frac{1}{3}\Big(m+k+c_1\Big)$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex-disjoint.

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arXiv2022

Relating dissociation, independence, and matchings

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is algorithmically hard even when $G$ is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple $\frac{4}{3}$-approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality ${\rm diss}(G)\leq\frac{4}{3}α(G-M)$ implicit in their work, where $G$ is a bipartite graph, $M$ is a maximum matching in $G$, and $α(G-M)$ denotes the independence number of $G-M$. We show that the pairs $(G,M)$ for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph $G$ satisfies $\max\{ α(G),2ν_s(G)\} \leq {\rm diss}(G)\leq α(G)+ν_s(G)\leq 2α(G)$, where $ν_s(G)$ denotes the induced matching number of $G$. We show that deciding whether ${\rm diss}(G)$ equals any of the four terms lower and upper bounding ${\rm diss}(G)$ is NP-hard.

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arXiv2021

Diameter, edge-connectivity, and $C_4$-freeness

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $d \leq (2n-3)/5$ for every connected $C_4$-free graph of order $n$, diameter $d$, and edge-connectivity at least $3$, which is best possible up to a small additive constant. For edge-connectivity at least $4$, we improve this to $d \leq (n-3)/3$. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power $q$ at least $7$, and every positive integer $k$, we show the existence of a connected $C_4$-free graph of order $n=(q^2+q-1)k+1$, diameter $d=4k$, and edge-connectivity $λ$ at least $q-6$, in particular, $d\geq 4(n-1)/(λ^2+O(λ))$.

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arXiv2021

Unbalanced spanning subgraphs in edge labeled complete graphs

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $Δ$. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(c_{d,Δ}-O\left(\frac{1}{n}\right)\right)m(G)$, where $c_{d,Δ}=d+Ω\left(\frac{1}Δ\right)$ for fixed $d$, that is, this number visibly deviates from its expected value when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $Δ\leq 2$, we present more detailed results.

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arXiv2021

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every $k$-edge colored color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$. For a perfect matching $M$ of $K_{2kn}$, a natural measure for the total deviation of $M$ from being color-balanced is $f(M)=\sum\limits_{i=1}^k|m_i(M)-n|$. While not every color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$, that is, a perfect matching with $f(M)=0$, we prove the existence of a perfect matching $M$ with $f(M)=O\left(k\sqrt{kn\ln(k)}\right)$ for general $k$ and $f(M)\leq 2$ for $k=3$; the case $k=2$ has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.

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arXiv2021

FPT algorithms for packing $k$-safe spanning rooted sub(di)graphs

We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing XP algorithms. The latter of these results answers a question of Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016]. Further, given an integer $k$, a graph $G=(V,E)$, and $r \in V$, we consider the problem of finding two arc-disjoint $(r,k)$-safe spanning trees. We show that this problem is also FPT with parameter $k$, again improving on a previous XP algorithm. Our main technical contribution is to prove that the existence of such spanning substructures is equivalent to the existence of substructures with size and maximum (out-)degree both bounded by a (linear or quadratic) function of $k$, which may be of independent interest.

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arXiv2021

Maximally distance-unbalanced trees

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness ${\rm uB}(G)$ of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. For positive integers $n$ up to $15$, they determine the trees $T$ of fixed order $n$ with the smallest and the largest values of ${\rm uB}(T)$, respectively. While the smallest value is achieved by the star $K_{1,n-1}$ for these $n$, which we then proved for general $n$ (Minimum distance-unbalancedness of trees, Journal of Mathematical Chemistry, DOI 10.1007/s10910-021-01228-4), the structure of the trees maximizing the distance-unbalancedness remained unclear. For $n$ up to $15$ at least, all these trees were subdivided stars. Contributing to problems posed by Miklavič and Šparl, we show $$\max\Big\{{\rm uB}(T):T\mbox{ is a tree of order }n\Big\} =\frac{n^3}{2}+o(n^3)$$ and $$\max\Big\{{\rm uB}(S(n_1,\ldots,n_k)):1+n_1+\cdots+n_k=n\Big\} =\left(\frac{1}{2}-\frac{5}{6k}+\frac{1}{3k^2}\right)n^3+O(kn^2),$$ where $S(n_1,\ldots,n_k)$ is the subdivided star such that removing its center vertex leaves paths of orders $n_1,\ldots,n_k$.

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arXiv2021

Efficiently finding low-sum copies of spanning forests in zero-sum complete graphs via conditional expectation

For a fixed positive $ε$, we show the existence of a constant $C_ε$ with the following property: Given a $\pm1$-edge-labeling $c:E(K_n)\to \{ -1,1\}$ of the complete graph $K_n$ with $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$ of maximum degree $Δ$, one can determine in polynomial time an isomorphic copy $F'$ of $F$ in $K_n$ with $|c(E(F'))|\leq \left(\frac{3}{4}+ε\right)Δ+C_ε.$ Our approach is based on the method of conditional expectation.

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