Search

Bollob\'as proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $\ell$ and $k$. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length $0 \bmod 4$.

The planar Tur\'an number ${\rm ex}_{\mathcal{P}}(n,C_k)$ is the largest number of edges in an $n$-vertex planar graph with no cycle of length $k$. Let $k\ge 11$ and $C,D$ be constants. Cranston, Lidick\'y, Liu and Shantanam \cite{2021Planar}, and independently Lan and Song \cite{LanSong} showed that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{Cn}{k}$ for large $n$. Moreover, Cranston et al. conjectured that ${\rm ex}_{\mathcal{P}}(n,C_k)\le 3n-6-\frac{Dn}{k^{lg_23}}$ when $n$ is large. In this note, we prove that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{6\cdot 3^{lg_23}n}{k^{lg_23}}$ for every $k$. It implies Cranston et al.'s conjecture is essentially best possible., Comment: 5 pages, 1 figures

In inverse optimization problems, the goal is to modify the costs in an underlying optimization problem in such a way that a given solution becomes optimal, while the difference between the new and the original cost functions, called the deviation vector, is minimized with respect to some objective function. The $\ell_1$- and $\ell_\infty$-norms are standard objectives used to measure the size of the deviation. Minimizing the $\ell_1$-norm is a natural way of keeping the total change of the cost function low, while the $\ell_\infty$-norm achieves the same goal coordinate-wise. Nevertheless, none of these objectives is suitable to provide a balanced or fair change of the costs. In this paper, we initiate the study of a new objective that measures the difference between the largest and the smallest weighted coordinates of the deviation vector, called the weighted span. We give a min-max characterization for the minimum weighted span of a feasible deviation vector, and provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights., Comment: 47 pages, 2 figures, 1 table

In minimum-cost inverse optimization problems, we are given a feasible solution to an underlying optimization problem together with a linear cost function, and the goal is to modify the costs by a small deviation vector so that the input solution becomes optimal. The difference between the new and the original cost functions can be measured in several ways. In this paper, we focus on two objectives: the weighted bottleneck Hamming distance and the weighted $\ell_\infty$-norm. We consider a general model in which the coordinates of the deviation vector are required to fall within given lower and upper bounds. For the weighted bottleneck Hamming distance objective, we present a simple, purely combinatorial algorithm that determines an optimal deviation vector in strongly polynomial time. For the weighted $\ell_\infty$-norm objective, we give a min-max characterization for the optimal solution, and provide a pseudo-polynomial algorithm for finding an optimal deviation vector that runs in strongly polynomial time in the case of unit weights. For both objectives, we assume that an algorithm with the same time complexity for solving the underlying combinatorial optimization problem is available. For both objectives, we also show how to extend the results to inverse optimization problems with multiple cost functions., Comment: 18 pages

We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding connectivity can be given. In this article, we investigate the problem of determining the maximum number of edges that can be removed from a color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, then we give a polynomial-time approximation algorithm for it. To analyze the approximation factor of this algorithm, we determine the minimum number of edges of color-avoiding connected graphs on a given number of vertices and with a given number of colors. Furthermore, we also consider a generalization of edge-color-avoiding connectivity to matroids.

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Tur\'an problem, was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path., Comment: 22 pages, 9 figures

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ if $H$ is an edge-critical graph or $C_4$ and asked if this equality holds for any graph $H$. All known exact values of this question require $H$ to contain at least one cycle. In this paper we focus on acyclic graphs and have the following results: (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double broom. (2) A \emph{tail} in $H$ is a path $P_3=v_0v_1v_2$ such that $v_2$ is only adjacent to $v_1$ and $v_1$ is only adjacent to $v_0,v_2$ in $H$. We obtain a tight upper bound for $f(n,H)$ when $H$ is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) Liu, Pikhurko and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ when $T$ is a tree. We provide an upper bound for $f_{2k}(n,P_{2k})$ and show it is tight when $2k-1$ is prime. This provides a negative answer to their question.

Katona and Varga showed that for any rational number $t \in (1/2,1]$, no chordal graph is minimally $t$-tough. We conjecture that no chordal graph is minimally $t$-tough for $t>1/2$ and prove several results supporting the conjecture. In particular, we show that for $t>1/2$, no strongly chordal graph is minimally $t$-tough, no split graph is minimally $t$-tough, and no chordal graph with a universal vertex is minimally $t$-tough.

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erd\H{o}s-R\'{e}nyi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size., Comment: 59 pages + Appendix + List of notation. Added reference to the recent arXiv preprint arXiv:2211.16086

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph.

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three., Comment: The paper has been expanded to include further results on paths

We introduce a new class of inverse optimization problems in which an input solution is given together with $k$ linear weight functions, and the goal is to modify the weights by the same deviation vector $p$ so that the input solution becomes optimal with respect to each of them, while minimizing $\|p\|_1$. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest $s$-$t$ path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min-max characterizations for the $\ell_1$-norm of an optimal deviation vector. Furthermore, we show that the optimal $p$ is not necessarily integral even when the weight functions are so, therefore computing an optimal solution is significantly more difficult than for the single-weighted case. We also give a necessary and sufficient condition for the existence of an optimal deviation vector that changes the values only on the elements of the input solution, thus giving a unified understanding of previous results on arborescences and matchings., Comment: 20 pages, 5 figures

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. The main results of this paper are the following. For any positive rational number $t \le 1$ and for any $k \ge 2$ and $r \ge 6$ integers recognizing $t$-tough bipartite graphs is coNP-complete (the case $t=1$ was already known), and this problem remains coNP-complete for $k$-connected bipartite graphs, and so does the problem of recognizing 1-tough r-regular bipartite graphs. To prove these statements we also deal with other related complexity problems on toughness. % In this paper we prove the following. For any positive rational number $t$, deciding whether $\tau(G)=t$ is DP-complete and if $t < 1$, this problem remains DP-complete for bipartite graphs. For any integer $k \ge 2$ and positive rational number $t \le 1$, recognizing $t$-tough $k$-connected bipartite graphs is coNP-complete. For any integer $r \ge 5$, recognizing $1/2$-tough $r$-regular graphs is coNP-complete. For any integer $r \ge 6$, recognizing 1-tough $r$-regular bipartite graphs is coNP-complete. For any positive rational number $t < 2/3$ we give a polynomial time algorithm for recognizing 3-regular graphs with toughness $t$. Finally, we prove that every connected 4-regular graph is 1/2-tough.

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity., Comment: 14 pages

An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for the decomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph.

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally $t$-tough graphs is DP-complete for any positive rational number $t$. We introduce a new notion called weighted toughness, which has a key role in our proof.

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in every minimally $1$-tough graph $\delta(G)\le\frac{n+2}{3}$. We also prove that every minimally $1$-tough claw-free graph is a cycle. On the other hand, we show that for every $t \in \mathbb{Q}$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.