arXiv subjects

Ingo Schiermeyer

Publications and source records attributed to Ingo Schiermeyer.

26 records · Page 2Linked to original sources

arXiv2018

Graphs with conflict-free connection number two

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 graph $G,$ let $C(G)$ be the subgraph of $G$ induced by its set of cut-edges. In this paper, we first show that, if $G$ is a connected non-complete graph $G$ of order $n\geq 9$ with $C(G)$ being a linear forest and with the minimum degree %$δ(G)\geq 2$, then $cfc(G)=2$ for $4 \leq n\leq 8 $; if $δ(G)\geq \max\{3, \frac{n-4}{5}\}$, then $cfc(G)=2$. The bound on the minimum degree is best possible. Next, we prove that, if $G$ is a connected non-complete graph of order $n\geq 33$ with $C(G)$ being a linear forest and with $d(x)+d(y)\geq \frac{2n-9}{5}$ for each pair of two nonadjacent vertices $x, y$ of $V(G)$, then $cfc(G)=2$. Both bounds, on the order $n$ and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on $G$ and the number of cut edges in $G$.

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arXiv2017

Maximum independent sets near the upper bound

The size of a largest independent set of vertices in a given graph $G$ is denoted by $α(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether $α(G) \geq K.$ An upper bound for the independence number $α(G)$ of a given graph $G$ with $n$ vertices and $m $ edges is given by $α(G) \leq p:=\lfloor\frac{1}{2} + \sqrt{\frac{1}{4} + n^2 - n - 2m}\rfloor.$ In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity $O(n^2)$ that, given as an input a graph $G$ with $n$ vertices, $m$ edges, $p:=\lfloor\frac{1}{2} + \sqrt{\frac{1}{4} + n^2 - n - 2m}\rfloor,$ and an integer $k \geq 0$ with $p \geq 2k+1,$ returns an induced subgraph $G_{p,k}$ of $G$ with $n_0 \leq p+2k+1$ vertices such that $α(G) \leq p-k$ if and only if $α(G_{p,k}) \leq p-k.$ Furthermore, we will show that we can decide in time $O(1.2738^{3k} + kn)$ whether $α(G_{p,k}) \leq p-k.$

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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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arXiv2015

On star-wheel Ramsey numbers

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\bar{G}$ contains a $G_2$. In this note, we determined the Ramsey number $R(K_{1,n},W_m)$ for even $m$ with $n+2\leq m\leq 2n-2$, where $W_m$ is the wheel on $m+1$ vertices, i.e., the graph obtained from a cycle $C_m$ by adding a vertex $v$ adjacent to all vertices of the $C_m$.

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arXiv2013

Graphs with $4$-rainbow index $3$ and $n-1$

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 called a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an {\it $S$-tree}. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow index} of $G$, denoted by $rx_k(G)$. Notice that an lower bound and an upper bound of the $k$-rainbow index of a graph with order $n$ is $k-1$ and $n-1$, respectively. Chartrand et al. got that the $k$-rainbow index of a tree with order $n$ is $n-1$ and the $k$-rainbow index of a unicyclic graph with order $n$ is $n-1$ or $n-2$. Li and Sun raised the open problem of characterizing the graphs of order $n$ with $rx_k(G)=n-1$ for $k\geq 3$. In early papers we characterized the graphs of order $n$ with 3-rainbow index 2 and $n-1$. In this paper, we focus on $k=4$, and characterize the graphs of order $n$ with 4-rainbow index 3 and $n-1$, respectively.

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arXiv2012

Rainbow numbers for graphs with cyclomatic number at most two

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 numbers f(n;H) and rb(Kn;H) are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph H with respect to its cyclomatic number. Let H be a graph of order p >= 4 and cyclomatic number v(H) >= 2: Then rb(Kn;H) cannot be bounded from above by a function which is linear in n: If H has cyclomatic number v(H) = 1; then rb(Kn;H) is linear in n: We will compute all rainbow numbers for the bull B; which is the unique graph with 5 vertices and degree sequence (1; 1; 2; 3; 3): Furthermore, we will compute some rainbow numbers for the diamond D = K_4 - e; for K_2;3; and for the house H (complement of P_5).

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arXiv2011

The rainbow connection number of 2-connected graphs

The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most the ceiling of n/2. The bound is optimal.

math.CO↗

arXiv2011

Rainbow connection number of dense graphs

An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.

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