On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

Online list coloring for signed graphs

We generalize the notion of online list coloring to signed graphs. We define the online list chromatic number of a signed graph, and prove a generalization of Brooks’ Theorem. We also give necessary and sufficient conditions for a signed graph to be degree paintable, or degree choosable. Finally, we classify the 2-list-colorable and 2-list-paintable signed graphs

Colorings of graphs, digraphs, and hypergraphs

Brooks' Theorem ist eines der bekanntesten Resultate über Graphenfärbungen: Sei G ein zusammenhängender Graph mit Maximalgrad d. Ist G kein vollständiger Graph, so lassen sich die Ecken von G so mit d Farben färben, dass zwei benachbarte Ecken unterschiedlich gefärbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem für Färbungen von Hypergraphen und gerichteten Graphen. Eine Färbung eines Hypergraphen ist eine Färbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. Kernstück ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,…,f_p) von Funktionen, welche von V(H) in die natürlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,…,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthält, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+…+f_p(v) \geq d_H(v) für alle v fast immer ausreichend für die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die Ausnahmefälle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-Färbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht Färbungen gerichteter Graphen. Eine azyklische Färbung eines gerichteten Graphen ist eine Färbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische Färbungsresultate übertragen. Dazu zählt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-Färbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-Färbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen für diese angegeben und die gerichtete Version des Satzes von Hajós bewiesen.Brooks‘ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,…,f_p) of functions, we examine if there is a partition of $H$ into induced subhypergraphs H_1,…,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v

Parameterized Inapproximability of Independent Set in HH-Free Graphs

We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halld\'orsson [SODA 1995] showed that for every $\delta>0$ IS has a polynomial-time $(\frac{d-1}{2}+\delta)$-approximation in $K_{1,d}$-free graphs. We extend this result by showing that $K_{a,b}$-free graphs admit a polynomial-time $O(\alpha(G)^{1-1/a})$-approximation, where $\alpha(G)$ is the size of a maximum independent set in $G$. Furthermore, we complement the result of Halld\'orsson by showing that for some $\gamma=\Theta(d/\log d),$ there is no polynomial-time $\gamma$-approximation for these graphs, unless NP = ZPP. Bonnet et al. [IPEC 2018] showed that IS parameterized by the size $k$ of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least $4$, (2) the star $K_{1,4}$, and (3) any tree with two vertices of degree at least $3$ at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken condition (2) that $G$ does not contain $K_{1,5}$). First, under the ETH, there is no $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Then, under the deterministic Gap-ETH, there is a constant $\delta>0$ such that no $\delta$-approximation can be computed in $f(k) \cdot n^{O(1)}$ time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime $f(k)\cdot n^{o(k)}$. Finally, we consider the parameterization by the excluded graph $H$, and show that under the ETH, IS has no $n^{o(\alpha(H))}$ algorithm in $H$-free graphs and under Gap-ETH there is no $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$.Comment: Preliminary version of the paper in WG 2020 proceeding

Weak degeneracy of graphs

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly $d$-degenerate, then $\chi(G) \leq d + 1$ (and, moreover, the same bound holds for the list-chromatic and even the DP-chromatic number of $G$). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly $4$-degenerate, which implies Thomassen's famous theorem that planar graphs are $5$-list-colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph $G$ of maximum degree $d \geq 3$ is weakly $(d-1)$-degenerate unless $G \cong K_{d + 1}$. (By contrast, all $d$-regular graphs have degeneracy $d$.) We actually prove an even stronger result, namely that for every $d \geq 3$, there is $\epsilon > 0$ such that if $G$ is a graph of weak degeneracy at least $d$, then either $G$ contains a $(d+1)$-clique or the maximum average degree of $G$ is at least $d + \epsilon$. Finally, we show that graphs of maximum degree $d$ and either of girth at least $5$ or of bounded chromatic number are weakly $(d - \Omega(\sqrt{d}))$-degenerate, which is best possible up to the value of the implied constant.Comment: 21 p