Ingo Schiermeyer

TU Bergakademie Freiberg · Institute of Discrete Mathematics and Algebra

Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 32nd edition of the workshop which took place in Poprad, Slovakia during September 8-13, 2024 (s...

For two graphs G and H, the Gallai–Ramsey number\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {gr}}_k(G:H)$$\end{document} is defined as the minimum integer n...

A compatible spanning circuit in an edge-colored graph G (not necessarily properly) is defined as a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. The existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and compatible Euler tours) has been studied exten...

The Ramsey number $r(G,H)$ is the minimum $N$ such that any red-blue coloring of the edges of the complete graph $K_{N}$ contains a blue copy of $G$ or a red copy of $H$. If both $G$ and $H$ are complete graphs $K_{m}$ and $K_{n}$, respectively, then we can restate the definition as the formulation that the Ramsey number $r(m,n)$ is the smallest $N...

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-colou...

Given a set H of graphs, let fH⋆:N>0→N>0 be the optimal χ-binding function of the class of H-free graphs, that is, fH⋆(ω)=max{χ(G):G is H-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 P5-free graphs and of (C5,C7,…)...

An edge-coloured graph G is called properly k-connected if any two vertices are connected by at least k internally vertex-disjoint paths whose edges are properly coloured. The proper k-connection number of a k-connected graph G, denoted by pck(G), is the minimum number of colours that are needed in order to make it properly k-connected. Let fk(n,r)...

A path in a vertex-coloured graph is called conflict-free if there is a colour used on exactly one of its vertices. A vertex-coloured graph is said to be conflict-free vertex-connected if any two distinct vertices of the graph are connected by at least one conflict-free vertex-path. The conflict-free vertex-connection number, denoted by vcfc(G), is...

Given a graph G and a positive integer k, define the Gallai—Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain exact values of the Gallai—Ramsey numbers for the union of two stars in many cases and...

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-col...

For every graph X, we consider the class of all connected {K1,3,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 P6, K1∪P5, 2P3, Z2 or K1∪Z1. Furthermore, for X chosen as 2K1∪K3, we list all eight imper...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargi...

Given a graph H $H$, the k $k$‐colored Gallai–Ramsey number g r k ( K 3 : H ) $g{r}_{k}({K}_{3}:H)$ is defined to be the minimum integer n $n$ such that every k $k$‐coloring of the edges of the complete graph on n $n$ vertices contains either a rainbow triangle or a monochromatic copy of H . $H.$ Fox et al. conjectured the values of the Gallai–Rams...

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant (CO-irredundant) if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ ($y\in V$) that is adjacent to $x$ and to no other vertex of $X$. The irredundant Ramsey number $s(t_{1},\ldots,t_{l})$, CO-irredunda...

Colourful connection concepts in graph theory such as rainbow connection, proper connection, odd connection or conflict-free connection have received a lot of attention. For an integer \(k \ge 1\) we call a path P in a graph G k-colourful, if at least k vertices of P are pairwise differently coloured. A graph G is k-colourful connected, if any two...

An edge-coloured graph G is called conflict-free connected if every two distinct vertices are connected by at least one path, which contains a colour used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is the smallest number of colours needed in order to make it conflict-free connected. F...

A path of a vertex-colored graph is conflict-free path, if there exists a color used only on one of its vertices; a vertex-colored graph is conflict-free vertex-connected, if there is a conflict-free path between each pair of distinct vertices of the graph. For a connected graph G, the minimum number of colors required to make G conflict-free verte...

Given a graph H, the k-colored Gallai-Ramsey number grk(K3:H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erdős-Hajnal conjecture for rainbow triangles. J. Combin....

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...

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai–Ramsey numbers for the graph G=Str obtaine...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the exact value of the Gallai-Ramsey numbers for...

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $\chi$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(\omega)=\max\{\chi(G): G\text{ is } \mathcal{H}\text{-free, } \omega(G)=\omega\}.$$ In this paper, we combine the two decomposition m...

A path in an edge-coloured graph is called a rainbow path if its edges receive pairwise distinct colours. An edge-coloured graph is said to be rainbow connected if any two distinct vertices of the graph are connected by a rainbow path. The minimum k for which there exists such an edge-colouring is the rainbow connection number rc(G) of G. Recently,...

Given a graph H, the k‐colored Gallai‐Ramsey number grk(K3⁢ :H) is defined to be the minimum integer n such that every k‐coloring (using all k colors) of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Recently, Fox et al [J. Combin. Theory Ser. B, 111 (2015), pp. 75–125] conjectured the value of th...

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The minimum $k$ for which there exists such an edge-colouring is the rainbow connection number $rc(G)$...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for the...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very...

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 $2\omega$-colourable where $\omega$ 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,...

Considering connected K 1,3-free graphs with independence number at least 3, Chud-novsky 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 d...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for fans Fm = K1 + mK2 and...

Given a graph H, the k-colored Gallai Ramsey number gr k (K3 : H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erd˝ os-Hajnal conjecture for rainbow triangles. J. C...

A graph G with clique number ω(G) and chromatic number χ(G) is perfect if χ(H) = ω(H) for every induced subgraph H of G. A family G of graphs is called χ-bounded with binding function f if χ(G′) ≤ f(ω(G′)) holds whenever G∈ G and G′ is an induced subgraph of G. In this paper we will present a survey on polynomial χ-binding functions. Especially we...

A path in a vertex-coloured graph is called \emph{conflict-free} if there is a colour used on exactly one of its vertices. A vertex-coloured graph is said to be \emph{conflict-free vertex-connected} if any two distinct vertices of the graph are connected by a conflict-free vertex-path. The \emph{conflict-free vertex-connection number}, denoted by $...

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and received a lot of attention. In this paper, we present a general concept of connection in graphs. As a particular case, we introduce the odd connection number and the odd vertex-connection number of a graph. Furthermore, we compute a...

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. For a...

In this paper, we study the chromatic number of [Formula presented]-free graphs. We show linear [Formula presented]-binding functions for several subclasses of [Formula presented]-free graphs, namely [Formula presented]-free graphs where [Formula presented]. We will also discuss [Formula presented]-binding functions for [Formula presented]-free gra...

Given two graphs G and H and a positive integer k, the k-color Gallai-Ramsey number, denoted by gr k (G : H), is the minimum integer N such that for all n ≥ N , every k-coloring of the edges of K n contains either a rainbow copy of G or a monochromatic copy of H. We prove that gr k (K 3 : C 2ℓ+1) = ℓ · 2 k + 1 for all k ≥ 1 and ℓ ≥ 3.

In this paper we continue studying the independent transversal domination number in a graph, i.e., the cardinality of a smallest dominating set which intersects each maximum independent set. As our main result, we prove two upper bounds on the independent transversal domination number in terms of the order and minimum degree of a graph.

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether $\alpha(G) \geq K.$ An upper bound for the independence number $\alpha(G)$ of a given graph $G$ with $n$ vertices and...

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether $\alpha(G) \geq K.$ An upper bound for the independence number $\alpha(G)$ of a given graph $G$ with $n$ vertices and...

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. For a...

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper, we consider sufficient conditions in terms of the ra...

We consider sequential heuristics methods for the Maximum Independent Set (MIS) problem. Three classical algorithms, VO [11], MIN [12], or MAX [6], are revisited. We combine Algorithm MIN with the α-redundant vertex technique [3]. Induced forbidden subgraph sets, under which the algorithms give maximum independent sets, are described. The Caro-Wei...

In this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G...

An edge-coloured graph. G is called. properly connected if any two vertices are connected by a path whose edges are properly coloured. The. proper connection number of a connected graph. G, denoted by. pc(G), is the smallest number of colours that are needed in order to make. G properly connected. Our main result is the following: Let. G be a conne...

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper we consider sufficient conditions in terms of connect...

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper we consider sufficient conditions in terms of the rat...

The weight of an edge of a graph is defined to be the sum of degrees of vertices incident to the edge. The weight of a graph G is the minimum of weights of edges of G. Jendrol’ and Schiermeyer (Combinatorica 21:351–359, 2001) determined the maximum weight of a graph having n vertices and m edges, thus solving a problem posed in 1990 by Erdős. The p...

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of G is the minimum number of colors that are needed in order to make G rainbow connected. In this paper, we complete the discussion of pairs (X,Y) of connecte...

An edge-coloured connected graph G is rainbow connected if each two vertices are connected by a path whose edges have distinct colours. If such a colouring uses k colours then G is called k-rainbow connected. The rainbow connection number of G, denoted by rc(G), is the minimum k such that G is k-rainbow connected. Even the problem to decide whether...

A vertex in a graph is called -redundant if , where stands for the independence number of , i.e. the maximum size of a subset of pairwise non-adjacent vertices. We will recall some results about -redundant vertices and show some new sufficient conditions for a vertex to be -redundant. Based on this, we will give a unified view about vertex removal...

The research in the present paper was motivated by the conjecture of Ryj\'{a}\v{c}ek 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 $G$...

Caro and Wei independently showed that the independence number α(G) of a graph G is at least ∑u∈V(G)1dG(u)+1. In the present paper we conjecture the stronger bound α(G)≥∑u∈V(G)2dG(u)+ωG(u)+1 where ωG(u) is the maximum order of a clique of G that contains the vertex u. We discuss the relation of our conjecture to recent conjectures and results conce...

The weight of an edge of a graph is defined to be the sum of degrees of vertices incident to the edge. The weight of a graph G is the minimum of weights of edges of G. Jendrol’ and Schiermeyer (Combinatorica 21:351–359, 2001) determined the maximum weight of a graph having n vertices and m edges, thus solving a problem posed in 1990 by Erdős. The p...

Let G be a nontrivial connected graph with an edge-coloring c: E(G) -> {1, 2, ..., q}, q is an element of N, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S subset of V(G), a tree that connects S in G is called an S-tree. The minimum number of color...

The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set of G. We also describe two modificat...

For a given graph H and n≥1, let f(n,H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n,H)=f(n,H)+1 colours contains a rainbow copy of H. The n...

A digraph is hamiltonian if it has a cycle that visits every vertex. If a digraph \(D\) is nonhamiltonian and \(D-v\) is hamiltonian for every \(v\in V(D)\) , then \(D\) is said to be hypohamiltonian. It is known that there exist hypohamiltonian digraphs of order \(n\) for every \(n\ge 6\) . Several infinite families of hypohamiltonian oriented gra...

For two given graphs (Formula presented.) and (Formula presented.), the Ramsey number (Formula presented.) is the least integer r such that for every graph G on r vertices, either G contains a (Formula presented.) or (Formula presented.) contains a (Formula presented.). In this note, we determined the Ramsey number (Formula presented.) for even m w...

Let G = (V (G), E(C)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) -> {1, 2, ... ,q}, q is an element of N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S subset of V(C), a tree connecting S in G is called an S-tree. The minim...

Abstract The Maximum Independent Set problem is NP-hard and remains NP-hard for graphs of maximum degree at most three (also called subcubic graphs). In this paper, we will study its complexity in subclasses of subcubic graphs. Let Si,j,k be the graph consisting of three induced paths of lengths i,j and k, with a common initial vertex and Alq be th...

A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of...

The weight of an edge xy of a graph G is deg G (x) + deg G (y) and the weight of G is the minimum over all weights of edges of G. The problem of finding the maximum weight of a graph having n vertices and m edges was posed by Erdős in 1990 and completely solved by Jen-drol' and Schiermeyer in 2001. This paper deals with a modification of the above...

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This graph parameter was introduced by G. Chartrand, G.L. Johns, MA. McKeon and P. Zhang in 200...

Given two graphs GG and HH, let f(G,H)f(G,H) denote the maximum number cc for which there is a way to color the edges of GG with cc colors such that every subgraph HH of GG has at least two edges of the same color. Equivalently, any edge-coloring of GG with at least rb(G,H)=f(G,H)+1rb(G,H)=f(G,H)+1 colors contains a rainbow copy of HH, where a rain...

Abstract A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant kF suc...

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This graph parameter was introduced by G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang in 20...

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a tree connecting $S$ in $G$ is called an $S$-tree. The minimum...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc (G), is the smallest number of colours that are needed in order to make G rainbow connected. M. Krivelevich and R. Yuster [J. Graph Theory 63, No. 3, 185–191 (2...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. A graph G is called rainbowk-connected if there is an edge colouring of G with k colours such that G is rainbow connected.In this paper we will study rainbow k-connected graphs with a minimum number of edges. For an integer n...

An edge-coloured connected graph G = (V, E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) <= k if vertical bar V(G)v...