Free Subgraph Research Articles

Single-exponential FPT algorithms for enumerating secluded [formula omitted]-free subgraphs and deleting to scattered graph classes

The celebrated notion of important separators bounds the number of small (S,T)-separators in a graph which is ‘farthest from S’ in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k. In this terminology, the bound on important separators says that there are at most 4k maximal k-secluded connected vertex sets C containing S but disjoint from T. We generalize this statement significantly: even when we demand that G[C] avoids a finite set F of forbidden induced subgraphs, the number of such maximal subgraphs is 2O(k) and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for Connectedk-SecludedF-Free Subgraph and for deleting into scattered graph classes.

A class of graphs of zero Turán density in a hypercube

AbstractFor a graph $H$ and a hypercube $Q_n$ , $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$ -free subgraph of $Q_n$ . If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$ , $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$ has a positive or zero Turán density remains a widely open question for general $H$ . By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of graphs, ones having so-called partite representation, that have zero Turán density. He asked whether this gives a characterisation, that is, whether a graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of graphs which have no partite representation, but on the other hand, have zero Turán density. In addition, we show that any graph whose every block has partite representation has zero Turán density in a hypercube.

Random polynomial graphs for random Turán problems

AbstractBukh and Conlon used random polynomial graphs to give effective lower bounds on , where is the th power of a balanced rooted tree . We extend their result to give effective lower bounds on , which is the maximum number of edges in a ‐free subgraph of the random graph . Analogous bounds for generalized Turán numbers in random graphs are also proven.

The feasibility problem for line graphs

We consider the following feasibility problem: given an integer n ≥ 1 and an integer m such that 0 ≤ m ≤ n 2 , does there exist a line graph L = L ( G ) with exactly n vertices and m edges? We say that a pair ( n , m ) is non-feasible if there exists no line graph L ( G ) on n vertices and m edges, otherwise we say ( n , m ) is a feasible pair. Our main result shows that for fixed n ≥ 5 , the values of m for which ( n , m ) is a non-feasible pair form disjoint blocks of consecutive integers which we determine completely. On the other hand we prove, among other things, that for the more general family of claw-free graphs (with no induced K 1 , 3 -free subgraph), all ( n , m ) -pairs in the range 0 ≤ m ≤ n 2 are feasible pairs.

LARGE -FREE SUBGRAPHS IN -CHROMATIC GRAPHS

AbstractFor a graph G and a family of graphs $\mathcal {F}$ , the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$ -free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math.342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.

Nordhaus–Gaddum problem in terms of G-free coloring

Let [Formula: see text] be a graph. A [Formula: see text]-coloring of [Formula: see text] is a mapping [Formula: see text], if each color class induces a [Formula: see text]-free subgraph. For a graph [Formula: see text] of order at least [Formula: see text], a [Formula: see text]-free [Formula: see text]-coloring of [Formula: see text], is a mapping [Formula: see text], so that the induced subgraph by each color class of [Formula: see text], contains no copy of [Formula: see text]. The [Formula: see text]-free chromatic number of [Formula: see text], is the minimum number [Formula: see text], so that it has a [Formula: see text]-free [Formula: see text]-coloring, and denoted by [Formula: see text]. Suppose that [Formula: see text] be a family of graphs, we say a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring, if there exists a map [Formula: see text], such that each color class [Formula: see text] does not contain any members of [Formula: see text]. In this paper, we give some bounds and attributes on the [Formula: see text]-free chromatic number of graphs in terms of the number of vertices, maximum degree, minimum degree, and chromatic number. Our main results are the Nordhaus–Gaddum-type theorem for the [Formula: see text]-free chromatic number of a graph.

Decomposition of 4k-regular graphs into k 4-regular K5-free and (K5 − e)-free subgraphs

Decomposition of 4k-regular graphs into k 4-regular K5-free and (K5 − e)-free subgraphs

Maximum $\mathcal{H}$-free subgraphs

Given a family of hypergraphs $\mathcal H$, let $f(m,\mathcal H)$ denote the largest size of an $\mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erd\H{o}s and Koml\'{o}s in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families $\{\mathcal H_m\}$ have bounded $f(m,\mathcal H_m)$ as $m\to\infty$? A variety of bounds for $f(m,\mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences $\{\mathcal H_m\}$ for which $f(m,\mathcal H_m)$ is bounded seems hopeless.

Improved bounds for the Erdős-Rogers function

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

Formula omitted]-free subgraphs of dense graphs maximizing the number of cliques and their blow-ups

We consider the structure of H-free subgraphs of graphs with high minimal degree. We prove that for every k>m there exists an ϵ≔ϵ(k,m)>0 so that the following holds. For every graph H with chromatic number k from which one can delete an edge and reduce the chromatic number, and for every graph G on n>n0(H) vertices in which all degrees are at least (1−ϵ)n, any subgraph of G which is H-free and contains the maximum number of copies of the complete graph Km is (k−1)-colorable.We also consider several extensions for the case of a general forbidden graph H of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs.

Many cliques in $H$-free subgraphs of random graphs

For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $\chi(H)-1$ partite subgraph of $G(n,p)$. The transition between these two behaviors occurs at $p=n^{-1/m_2(H)}$. When $m_2(H) 0$ small at $p=n^{-1/m_2(H)+\delta}$ one can typically still keep most of the copies of $K_m$ in an $H$-free subgraph of $G(n,p)$. Thus, the transition between the two behaviors in this case occurs at some $p$ significantly bigger than $n^{-1/m_2(H)}$. To show that the second case is not redundant we present a construction which may be of independent interest. For each $k \geq 4$ we construct a family of $k$ chromatic graphs $G(k,\epsilon_i)$ where $m_2(G(k,\epsilon_i))$ tends to $\frac{(k+1)(k-2)}{2(k-1)} (< m_2(K_{k-1}))$ as $i$ tends to infinity. This is tight for all values of $k$

Matching Trees for Simplicial Complexes and Homotopy Type of Devoid Complexes of Graphs

We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes. We then exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, $\mathcal {D}(G;\mathcal {F})$ whose faces are vertex subsets of G that induce $\mathcal {F}$ -free subgraphs, where G is a multigraph and $\mathcal {F}$ is a family of multigraphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.

Bounding the CP-rank by graph parameters

The cp-rank of a graph G, cpr(G), is the maximum cp-rank of a completely positive matrix with graph G. One obvious lower bound on cpr(G) is the (edge-) clique covering number, cc(G), i.e., the minimal number of cliques needed to cover all of G’s edges. It is shown here that for a connected graph G, cpr(G) = cc(G) if and only if G is triangle free and not a tree. Another lower bound for cpr(G) is tf(G), the maximum size of a triangle free subgraph of G. We consider the question of when does the equality cpr(G) = tf(G) hold.

Large Subgraphs without Short Cycles

We study two extremal problems about subgraphs excluding a family $\F$ of graphs. i) Among all graphs with $m$ edges, what is the smallest size $f(m,\F)$ of a largest $\F$--free subgraph? ii) Among all graphs with minimum degree $\delta$ and maximum degree $\Delta$, what is the smallest minimum degree $h(\delta,\Delta,\F)$ of a spanning $\F$--free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where $\F$ is composed of all even cycles of length at most $2r$, $r\geq 2$. In this case, we give bounds on $f(m,\F)$ and $h(\delta,\Delta,\F)$ that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph $G$, we show the existence of subgraphs with arbitrarily high girth, and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.

Decomposition of bounded degree graphs into [formula omitted]-free subgraphs

We prove that every graph with maximum degree Δ admits a partition of its edges into O(Δ) parts (as Δ→∞) none of which contains C4 as a subgraph. This bound is sharp up to a constant factor. Our proof uses an iterated random colouring procedure.

On the NP-completeness of the perfect matching free subgraph problem

Given a bipartite graph G=(U∪V,E) such that ∣U∣=∣V∣ and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U, either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arises in the structural analysis of differential-algebraic systems. The purpose of this paper is to show that this problem is NP-complete. We show that the problem is equivalent to the stable set problem in a restricted case of tripartite graphs. Then we show that the latter remains NP-complete in that case. We also prove the NP-completeness of the related minimum blocker problem in bipartite graphs with perfect matching.

On even-cycle-free subgraphs of the hypercube

It is shown that the size of any C 4 k + 2 -free subgraph of the hypercube Q n , k ⩾ 3 , is o ( e ( Q n ) ) .

Line-Polar Graphs: Characterization and Recognition

A graph is polar if its vertex set can be partitioned into and in such a way that induces a complete multipartite graph and induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite, cobipartite, and split graphs. The problem of recognizing polar graphs is NP-complete in general. However, it has been shown to be polynomial for several classes of graphs, including cographs and chordal graphs. In this paper, we study the problem of recognizing graphs whose line graphs are polar. It turns out that the core part of this problem lies in determining whether the edge set of a graph admits a partition so that induces a -free subgraph (i.e., a matching) and induces a -free subgraph. We give a structural characterization of such graphs. The characterization enables us to devise an time algorithm to solve the stated recognition problem.

Structural Analysis for Differential-Algebraic Systems: Complexity, Formulation and Facets

Abstract In this paper we consider the structural analysis problem for differential-algebraic systems with conditional equations. This consists, given a conditional differential algebraic system, in verifying if the system is well-constrained for every state and if not in finding a state for which the system is bad-constrained. We first show that the problem reduces to the perfect matching free subgraph problem in a bipartite graph. We then show the NP-completeness of this problem and give a formulation as an integer linear program. We also discuss the polytope of the solutions of this problem and propose a Branch-and-Cut algorithm.

Turán’s Theorem in the Hypercube

We are motivated by the analogue of Tura´n’s theorem in the hypercube $Q_n$: How many edges can a $Q_d$-free subgraph of $Q_n$ have? We study this question through its Ramsey-type variant and obtain asymptotic results. We show that for every odd $d$ it is possible to color the edges of $Q_n$ with $\frac{(d+1)^2}{4}$ colors such that each subcube $Q_d$ is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with ${d+1\choose 2} +1$ colors is not possible. The corresponding question for vertices is also considered. It is not possible to color the vertices of $Q_n$ with $d+2$ colors such that any $Q_d$ is polychromatic, but there is a simple $d+1$ coloring with this property. A relationship to anti-Ramsey colorings is also discussed. We discover much less about the Tura´n-type question which motivated our investigations. Numerous problems and conjectures are raised.