Gallai Theorem Research Articles

A localized approach for Turán number of long cycles

AbstractLet be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by over all edges in an ‐vertex graph is at most , and characterize all ‐vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.

The Maximum Number of Cliques in Graphs with Bounded Odd Circumference

AbstractIn this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

On the vertex stability numbers of graphs

For an arbitrary invariant ρ(G) of a graph G the ρ-vertex stability number vsρ(G) (ρ-edge stability number esρ(G)) is the minimum number of vertices (edges) whose removal results in a graph H⊆G with ρ(H)≠ρ(G). If such a vertex set (edge set) does not exist, then we set vsρ(G)=∞ (esρ(G)=∞).In the first part of this paper we give some general results for the ρ-vertex stability number. We prove among others a result which implies the well-known Theorem of Gallai on independence and vertex covering of graphs. In the second part we focus on the invariant chromatic number χ(G) and determine vsχ(G) for several classes of graphs. Moreover, we consider the relationship between esχ(G) and vsχ(G) and prove that their difference and their ratio may get arbitrarily large.

Solution to a Forcible Version of a Graphic Sequence Problem

Let \(A_n=(a_1,a_2,\ldots ,a_n)\) and \(B_n=(b_1,b_2,\ldots ,b_n)\) be nonnegative integer sequences with \(A_n\le B_n\). The purpose of this note is to give a good characterization such that every integer sequence \(\pi =(d_1,d_2,\ldots d_n)\) with even sum and \(A_n\le \pi \le B_n\) is graphic. This solves a forcible version of problem posed by Niessen and generalizes the Erdős–Gallai theorem.

A Sylvester–Gallai theorem for cubic curves

We prove a variant of the Sylvester–Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in R2 is not contained in a cubic, then there is a cubic that contains exactly nine of the points. This resolves the first unknown case of a conjecture of Wiseman and Wilson from 1988, who proved a variant of Sylvester–Gallai for conics and conjectured that similar statements hold for curves of any degree.

A generalized Sylvester–Gallai-type theorem for quadratic polynomials

Abstract In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$ circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials ${\mathcal {Q}}$ satisfies that for every two polynomials $Q_1,Q_2\in {\mathcal {Q}}$ there is a subset ${\mathcal {K}}\subset {\mathcal {Q}}$ such that $Q_1,Q_2 \notin {\mathcal {K}}$ and whenever $Q_1$ and $Q_2$ vanish, then $\prod _{i\in {\mathcal {K}}} Q_i$ vanishes, then the linear span of the polynomials in ${\mathcal {Q}}$ has dimension $O(1)$ . This extends the earlier result [21] that holds for the case $|{\mathcal {K}}| = 1$ .

Cover and variable degeneracy

Let f be a nonnegative integer valued function on the vertex set of a graph. A graph is strictlyf-degenerate if each nonempty subgraph Γ has a vertex v such that degΓ(v)<f(v). In this paper, we define a new concept, strictly f-degenerate transversal, which generalizes list coloring, signed coloring, DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. A cover of a graph G is a graph H with vertex set V(H)=⋃v∈V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; the edge set M=⋃uv∈E(G)Muv, where Muv is a matching between Xu and Xv. A vertex set R⊆V(H) is a transversal of H if |R∩Xv|=1 for each v∈V(G). A transversal R is a strictlyf-degenerate transversal if H[R] is strictly f-degenerate. The main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable result, signed degree-colorable result, and DP-degree-colorable result. We also give some structural results on critical graphs with respect to strictly f-degenerate transversal. Using these results, we can uniformly prove many new and known results. In the final section, we pose some open problems.

Real and complex supersolvable line arrangements in the projective plane

If we regard a set of s lines in \({\mathbb P}^2\) over either the reals or the complex numbers as an algebraic plane curve, then it is an open problem to classify for all s those for which the number \(t_2\) of points of multiplicity 2 satisfies \(t_2<\lfloor s/2\rfloor \). By the Sylvester–Gallai theorem, there are no nontrivial (i.e., not a pencil or a near pencil) real arrangements with \(t_2=0\), but there are complex arrangements with \(t_2=0\) and it is an open problem to classify them. In this paper, we initiate a classification of an interesting class of line arrangements called the supersovable line arrangements and give a partial classification for them over the reals or the complex numbers. In particular, we show that a complex line arrangement which is nontrivial cannot have more than 4 modular points and we completely describe those with 3 or 4 modular points.

An analogue of the Erdős–Gallai theorem for random graphs

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős–Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a Pn-free subgraph of G(N,p), practically for all values of N,n and p. Our work is also motivated by the recent progress on the size-Ramsey number of paths.

Double Points of Free Projective Line Arrangements

Abstract We prove the Anzis–Tohăneanu conjecture, that is, the Dirac–Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point that can be regarded as the Sylvester–Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also, we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

The formula for Turán number of spanning linear forests

Let F be a family of graphs. The Turán number ex(n;F) is defined to be the maximum number of edges in a graph of order n that is F-free. In 1959, Erdős and Gallai determined the Turán number of Mk+1 (a matching of size k+1) as follows: ex(n;Mk+1)=max2k+12,n2−n−k2.Since then, there has been a lot of research on Turán number of linear forests.A linear forest is a graph whose connected components are all paths or isolated vertices. Let Ln,k be the family of all linear forests of order n with k edges. In this paper, we prove that ex(n;Ln,k)=maxk2,n2−n−k−122+c,where c=0 if k is odd and c=1 otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in Wang and Yang (2019) as special cases.Moreover, we show that our main theorem implies Erdős–Gallai Theorem and also gives a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős Matching Conjecture.

Extensions of the Erdős–Gallai theorem and Luo’s theorem

AbstractThe famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

Erdős–Gallai stability theorem for linear forests

The Erdős–Gallai Theorem states that every graph of average degree more than l−2 contains a path of order l for l≥2. In this paper, we obtain a stability version of the Erdős–Gallai Theorem in terms of minimum degree. Let G be a connected graph of order n and F=(⋃i=1kP2ai)⋃(⋃i=1lP2bi+1) be k+l disjoint paths of order 2a1,…,2ak,2b1+1,…,2bl+1, respectively, where k≥0, 0≤l≤2, and k+l≥2. If the minimum degree δ(G)≥∑i=1kai+∑i=1lbi−1, then F⊆G except several classes of graphs for sufficiently large n, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.

A Characterization of Box-bounded Degree Sequences of Graphs

Let $$A_n=(a_1,a_2,\ldots ,a_n)$$ and $$B_n=(b_1,b_2,\ldots ,b_n)$$ be nonnegative integer sequences with $$A_n\le B_n$$ and $$b_i\ge b_{i+1},a_i+b_i\ge a_{i+1}+b_{i+1}, i=1,2\ldots , n-1$$ . The purpose of this note is to give a good characterization for $$A_n$$ and $$B_n$$ such that every integer sequence $$\pi =(d_1,d_2,\ldots d_n)$$ with even sum and $$A_n\le \pi \le B_n$$ is graphic. This improves related results of Guo and Yin and generalizes the Erdős–Gallai theorem.

Stability in the Erdős–Gallai Theorem on cycles and paths, II

The Erdős–Gallai Theorem states that for k≥3, any n-vertex graph with no cycle of length at least k has at most 12(k−1)(n−1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)≤max{h(n,k,2),h(n,k,⌊k−12⌋)}, where h(n,k,a)≔k−a2+a(n−k+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,⌊k−12⌋) edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for k≥3 odd and all n≥k, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)≤max{h(n,k,3),h(n,k,k−32)}. The upper bound for e(G) here is tight.

On extremal hypergraphs for forests of tight paths

In this paper, we investigate the maximal size of a k-uniform hypergraph containing no forests of tight paths, which extends the classical Erdős–Gallai Theorem for paths in graphs. Our results build on the results of Györi, Katona and Lemons, who considered the maximal size of a k-uniform hypergraph containing no single tight path.

Packing $k$-Matchings and $k$-Critical Graphs

A (simple) $k$-matching in a graph is a subgraph all of whose nodes have degree at most $k$. The $k$-matching problem is to find a $k$-matching with a maximum number of edges. Well-known results for the classical $1$-matching problem, such as the Tutte--Berge min-max theorem and Edmonds's polynomial-time algorithm, are known to generalize to $k$-matchings. These 1-matching results are also known to generalize to the problem of finding a subgraph, with a maximum number of nodes, whose connected components are isolated edges and 1-critical (or hypomatchable) graphs from a specified set (e.g., triangles or pentagons). In this paper we present a new common generalization of these two generalized 1-matching problems. We present an algorithm, a min-max theorem, and a structure theorem (that generalizes the Edmonds--Gallai theorem for 1-matchings). Even when specialized to $k$-matchings, our min-max and structure theorems appear to be new.

The maximum number of cliques in graphs without long cycles

The Erdős–Gallai Theorem states that for k≥3 every graph on n vertices with more than 12(k−1)(n−1) edges contains a cycle of length at least k. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples Hn,k,t and Hn,k,2. In this note, we generalize the result of Kopylov to bound the number of s-cliques in a graph with circumference less than k. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of s-cliques in a graph with no path on k-vertices.

Tight cycles and regular slices in dense hypergraphs

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results.Firstly, we prove a hypergraph extension of the Erdős–Gallai Theorem: for every δ>0 every sufficiently large k-uniform hypergraph with at least (α+δ)(nk) edges contains a tight cycle of length αn for each α∈[0,1].Secondly, we find (asymptotically) the minimum codegree requirement for a k-uniform k-partite hypergraph, each of whose parts has n vertices, to contain a tight cycle of length αkn, for each 0<α<1.

On the Sylvester–Gallai theorem for conics

In the present note we give a new proof of a result due to Wiseman and Wilson [13] which establishes an analogue of the Sylvester–Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specically, we use Cremona transformation of the projective plane and Hirzebruch inequality (1).