arXiv subjects

Christoph Brause

Publications and source records attributed to Christoph Brause.

11 recordsLinked to original sources

arXiv2025

Majority additive coloring and the maximum degree

Kamyczura introduced the notion of a majority additive $k$-coloring of a graph $G$ as a function $c: V(G) \to \{1,2,\ldots,k\}$ such that $$\left|\left\{u \in N_G(v):\sum_{w \in N_G(u)} c(w) = s \right\}\right|\leq \max\left\{1,\frac{d_G(v)}{2}\right\}$$ for every vertex $v$ of $G$ and every positive integer $s$. We show that every graph $G$ of maximum degree $Δ$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(Δ^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $Δ$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\geq 2$.

math.CO↗

arXiv2025

On the distinguishing chromatic number in hereditary graph classes

The distinguishing chromatic number of a graph $G$, denoted $χ_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $χ_D(G)\le 2Δ(G)$ for any connected graph $G$, and the equality holds for complete balanced bipartite graphs $K_{p,p}$ and for $C_6$. In this paper, we show that the upper bound on $χ_D(G)$ can be substantially reduced if we forbid some small graphs as induced subgraphs of $G$, that is, we study the distinguishing chromatic number in some hereditary graph classes.

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arXiv2024

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

Given a graph $G$ and sets $\{α_v~|~v \in V(G)\}$ and $\{β_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $α_v \le d_T (v) \le β_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 $α_v \le d_T (v) \le β_v $ for all $v \in U$ can be found in polynomial time if such a tree exists.

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arXiv2022

Loose edge-connection of graphs

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured graph $G$ is loose edge-connected if between any two of its vertices there is a path of length one, or a bi-coloured path of length two, or a path of length at least three with at least three colours used on its edges. The minimum number of colours, used in a loose edge-colouring of $G$, is called the loose edge-connection number and denoted $\lec(G)$. We determine the precise value of this parameter for any simple graph $G$ of diameter at least 3. We show that deciding, whether $\lec(G) = 2$ for graphs $G$ of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs $K_{r,s}$ with $\lec(K_{r,s}) = 2$.

math.CO↗

arXiv2022

Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $χ$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(ω)=\max\{χ(G): G\text{ is } \mathcal{H}\text{-free, } ω(G)=ω\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $χ$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $ω\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(ω)=f_{3K_1}^\star(ω)\in Θ(ω^2/\log(ω)),$ (ii) $\ f_{\{P_5,co-banner\}}^\star(ω)=f^\star_{\{2K_2\}}(ω)\in\mathcal{O}(ω^2),$ (iii) $\ f_{\{C_5,C_7,\ldots,banner\}}^\star(ω)=f^\star_{\{C_5,3K_1\}}(ω)\notin \mathcal{O}(ω),$ and (iv) $\ f_{\{P_5,C_4\}}^\star(ω)=\lceil(5ω-1)/4\rceil.$ We also characterise, for each of our considered graph classes, all graphs $G$ with $χ(G)>χ(G-u)$ for each $u\in V(G)$. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for $(P_5,banner)$-free graphs.

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arXiv2022

Partitioning H-Free Graphs of Bounded Diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to $H$-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

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arXiv2021

Acyclic, Star, and Injective Colouring: Bounding the Diameter

We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 4$, and Star $3$-Colouring is polynomial-time solvable if $d\leq 3$ but NP-complete for $d\geq 8$. As far as we are aware, Star $3$-Colouring is the first problem that exhibits a complexity jump for some $d\geq 3$. Our third main result is that $L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.

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arXiv2021

Forbidden induced subgraphs for perfectness of claw-free graphs of independence number at least 4

For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$ is an induced subgraph of some of $P_6$, $K_1 \cup P_5$, $2P_3$, $Z_2$ or $K_1 \cup Z_1$. Furthermore, for $X$ chosen as $2K_1 \cup K_3$, we list all eight imperfect graphs belonging to the class; and for every other choice of $X$, we show that there are infinitely many such graphs. In addition, for $X$ chosen as $B_{1,2}$, we describe the structure of all imperfect graphs in the class.

math.CO↗

arXiv2019

On forbidden induced subgraphs for K_{1,3}-free perfect graphs

Considering connected $K_{1,3}$-free graphs with independence number at least $3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is $2ω$-colourable where $ω$ denotes the clique number of $G$. We study $(K_{1,3}, Y)$-free graphs, and show that the following three statements are equivalent. (1) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is perfect. (2) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is $ω$-colourable. (3) $Y$ is isomorphic to an induced subgraph of $P_5$ or $Z_2$ (where $Z_2$ is also known as hammer). Furthermore, for connected $(K_{1,3}, Y)$-free graphs (without an assumption on the independence number), we show a similar characterisation featuring the graphs $P_4$ and $Z_1$ (where $Z_1$ is also known as paw).

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arXiv2017

A characterization of trees with equal 2-domination and 2-independence numbers

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $γ_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $α_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.

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arXiv2015

Local Connectivity, Local Degree Conditions, some Forbidden Induced Subgraphs, and Cycle Extendability

The research in the present paper was motivated by the conjecture of Ryjáček that every locally connected graph is weakly pancyclic. For a connected locally connected graph $G$ of order at least $3$, our results are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is weakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycle extendable if and only if $2δ(G)\geq n(G)$. If $G$ is $\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$-free or $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$-free, then $G$ is fully cycle extendable. If $G$ is distinct from $K_1+K_1+\bar{K}_3$ and $\{ K_1+P_4,K_{1,4},K_2+(K_1\cup K_2)\}$-free, then $G$ is fully cycle extendable. Furthermore, if $G$ is a connected graph of order at least $3$ such that $$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$ for every induced path $vuw$ of order $3$ in $G$, then $G$ is fully cycle extendable, which implies that every connected locally Ore or locally Dirac graph of order at least $3$ is fully cycle extendable.

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