Triangle-free Research Articles

The BCH Family of Storage Codes on Triangle-Free Graphs and Its Relation to R(3,t)

The BCH Family of Storage Codes on Triangle-Free Graphs and Its Relation to R(3,t)

Paired coalition in graphs

A paired coalition in a graph G = ( V , E ) consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a paired dominating set but whose union V 1 ∪ V 2 is a paired dominating set. A paired coalition partition (abbreviated pc-partition) in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that each set V i of π is not a paired dominating set but forms a paired coalition with another set V j ∈ π . The paired coalition graph PCG ( G , π ) of the graph G with the pc-partition π of G, is the graph whose vertices correspond to the sets of π , and two vertices V i and V j are adjacent in PCG ( G , π ) if and only if their corresponding sets V i and V j form a paired coalition in G. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G of order n with PC ( G ) = n and unicyclic graphs G of order n with PC ( G ) = n − 2 .

Chen-Chvátal’s conjecture for graphs with restricted girth

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Maximum size of a triangle-free graph with bounded maximum degree and matching number

Maximum size of a triangle-free graph with bounded maximum degree and matching number

A bound on relative lengths of triangle-free graphs

For a 2-connected graph G, the relative length of G, denoted by diff(G), is the difference between the orders of a longest path and a longest cycle in G. This parameter is used as a measure to estimate how close a given graph is to a hamiltonian graph. Let σk(G) be the least value of the sums of degrees of vertices in independent sets of cardinality k. In 2008, Paulusma and Yoshimoto proved that a 2-connected triangle-free graph G of order n with σ4(G)≥n+2 satisfies diff(G)≤1 unless G is isomorphic to one exceptional graph G0. In this paper, we extend their result and prove that for an integer s with 23(n+4)<s≤n+2, a 2-connected triangle-free graph of order n with σ4(G)≥s satisfies diff(G)≤n+3−s unless s=σ4(G)=n+2 and G=G0.

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Observations on graph invariants with the Lovász $ \vartheta $-function

&lt;abstract&gt;&lt;p&gt;This paper delves into three research directions, leveraging the Lovász $ \vartheta $-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that $ n \geq 14 $, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $ n $ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász $ \vartheta $-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the $ \vartheta $-function by Schrijver and the identical function by McEliece &lt;italic&gt;et al.&lt;/italic&gt; (1978). It shows, by a counterexample, that the $ \vartheta $-function variant by Schrijver does not possess the property of the Lovász $ \vartheta $-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory.&lt;/p&gt;&lt;/abstract&gt;

Image Classification of Leukemic Cells Using Invariants of Triangle-Free Graphs as Synthetic Features

Image Classification of Leukemic Cells Using Invariants of Triangle-Free Graphs as Synthetic Features

Solution to a problem of Katona on counting cliques of weighted graphs

A subset I of the vertex set V(G) of a graph G is called a k-clique independent set ofG if no k vertices in I form a k-clique of G. An independent set is a 2-clique independent set. Let πk(G) denote the number of k-cliques of G. For a function w:V(G)→{0,1,2,…}, let G(w) be the graph obtained from G by replacing each vertex v by a w(v)-clique Kv and making each vertex of Ku adjacent to each vertex of Kv for each edge {u,v} of G. For an integer m≥1, consider any w with ∑v∈V(G)w(v)=m. For U⊆V(G), we say that w is uniform onU if w(v)=0 for each v∈V(G)∖U and, for each u∈U, w(u)=m/|U| or w(u)=m/|U|. Katona asked if πk(G(w)) is smallest when w is uniform on a largest k-clique independent set of G. He placed particular emphasis on the Sperner graph Bn, given by V(Bn)={X:X⊆{1,…,n}} and E(Bn)={{X,Y}:X⊊Y∈V(Bn)}. He provided an affirmative answer for k=2 (and any G). We determine graphs for which the answer is negative for every k≥3. These include Bn for n≥2. Generalizing Sperner’s Theorem and a recent result of Qian, Engel and Xu, we show that πk(Bn(w)) is smallest when w is uniform on a largest independent set of Bn. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.

Advice complexity of adaptive priority algorithms

The priority model was introduced to capture “greedy-like” algorithms. Motivated by the success of advice complexity in the area of online algorithms, the fixed priority model was extended to include advice, and a reduction-based framework was developed for proving lower bounds on the amount of advice required to achieve certain approximation ratios in this rather powerful model. To capture most of the algorithms that are considered greedy-like, the even stronger model of adaptive priority algorithms is needed. We extend the adaptive priority model to include advice. We modify the reduction-based framework from the fixed priority case to work with the more powerful adaptive priority algorithms, simplifying the proof of correctness and strengthening all previous lower bounds by a factor of two in the process.As evidence that adding advice to adaptive priority algorithms extends both adaptive priority algorithms and online algorithms with advice, we present a purely combinatorial adaptive priority algorithm with advice for Minimum Vertex Cover on triangle-free graphs of maximum degree three. Our algorithm achieves optimality and uses at most 7n/22 bits of advice. No adaptive priority algorithm without advice can achieve optimality without advice, and we prove that an online algorithm with advice needs more than 7n/22 bits of advice to reach optimality.We show connections between exact algorithms and priority algorithms with advice. The branching in branch-and-reduce algorithms can be seen as trying all possible advice strings, and all priority algorithms with advice that achieve optimality define corresponding exact algorithms, priority exact algorithms. Lower bounds on advice-based adaptive algorithms imply lower bounds on running times of exact algorithms designed in this way.

Three-coloring triangle-free graphs on surfaces VI. 3-colorability of quadrangulations

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

We give an approximation algorithm for Quantum Max-Cut which works by rounding an SDP relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved approximation ratios of 0.531 and 0.533 respectively. In addition, we study the EPR Hamiltonian, which we argue is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio 1/2 on all graphs.

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.For a graph G, let IG denote its edge ideal, the monomial ideal generated by xixj for every edge ij of G. In this paper we study the graded Betti numbers of IG, which are combinatorial invariants that measure the complexity of a minimal free resolution of IG. The Betti numbers of the form βi,2i+2 constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain “parabola” inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let βi,j be a parabolic Betti number on the r-th row of the Betti table, for r≥3. We prove that almost all graphs G with βi,j(IG)=0 can be partitioned into r−2 cliques and one independent set. In particular, for almost all graphs G with βi,j(IG)=0, the regularity of IG is r−1.

Reconstruction and edge reconstruction of triangle-free graphs

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs {G−v:v∈V(G)}. Let diam(G) and κ(G) denote the diameter and the connectivity of a graph G, respectively, and let G2:={G:diam(G)=2} and G3:={G:diam(G)=diam(G‾)=3}. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in G2∪G3. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph G in G2∪G3 with κ(G)=2. Moreover, they asked whether the result still holds if κ(G)≥3. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in G2∪G3 which contain triangles.) The case when G∈G3 and κ(G)≥3 was recently confirmed by Devi Priya and Monikandan. In this paper, we further show the Reconstruction Conjecture holds for every triangle-free graph G in G2 with κ(G)=3. We also prove similar results about the Edge Reconstruction Conjecture.

10 problems for partitions of triangle-free graphs

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős’ Sparse Half Conjecture.Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition V(G)=A∪B with |A|=|B|=n/2 such that e(G[A])+e(G[B])≤n2/16. This result is sharp since the complete bipartite graph with class sizes 3n/4 and n/4 achieves equality, when n is a multiple of 4.Additionally, we discuss similar problems for K4-free graphs.

Burling graphs revisited, part II: Structure

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Any induced subgraph of a graph in this sequence is called a Burling graph. These graphs have attracted some attention because on one hand they have geometric representations, and on the other hand they provide counter-examples to several conjectures about bounding the chromatic number in classes of graphs.Using an equivalent definition from the first part of this work (called derived graphs), we study several structural properties of Burling graphs. In particular, we give decomposition theorems for the class using in-star cutsets, study holes and their interactions in Burling graphs, and analyze the effect of subdividing some arcs of a Burling graph. Using mentioned results, we introduce new techniques for providing new triangle-free graph that are not Burling graphs.Among other applications, we prove that wheels are not Burling graphs. This answers an open problem of the second author and disproves a conjecture of Scott and Seymour.

Burling graphs revisited, part III: Applications to [formula omitted]-boundedness

The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence.In the first and second parts of this work, we introduced derived graphs, a class of graphs, equal to the class of Burling graphs, and proved several geometric and structural results about them.In this third part, we use those results to find some Burling and non-Burling graphs, and we see some applications of this in the theory of χ-boundedness. Specifically, we show that several graphs, like K5, some series-parallel graphs that we call necklaces, and some other graphs are not weakly pervasive.

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S⊆V is called a g-extra cut of G if the graph induced by the set G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G, denoted as κg(G), is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ(G), also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ(G) and the graph G. In addition, we show that κ3(μ(G))=2κ1(G)+1 for κ1(G)≤min{4,⌊n2⌋}, and prove the bounds of κ2g+1(μ(G)) for g≥2.

Constructing extremal triangle-free graphs using integer programming

The maximum number of edges in a graph with matching number m and maximum degree d has been determined in Chvátal and Hanson (1976) and Balachandran and Khare (2009), where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In Ahanjideh et al. (2022), the problem is solved in triangle-free graphs for d≥m, and for d<m with either Z(d)≤m<2d or d≤6, where Z(d) is approximately 5d/4. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into d≤10 instead of d≤6 for m>d. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in Ahanjideh et al. (2022).

The exponential growth of the packing chromatic number of iterated Mycielskians

For a graph G, the packing chromatic number of G, denoted by χρ(G), is the smallest integer k such that there exists a coloring f:V(G)↦{1,…,k} of G where every two distinct vertices u and v such that f(u)=f(v) are at pairwise distance at least f(u)+1. In this paper, we study graphs G for which χρ(μt(G))=2tχρ(G) for all t≥1, where μt(G) stands for the t-iterated Mycielskian of G. We show that a first natural upper bound of χρ(μt(G)) is 2t(|V(G)|−α(G)+1) for any graph G where α(G) is the independence number of G. This bound is rounded to 2tχρ(G) if the diameter of G is two. If moreover the graph G belongs to P, class of graphs whose vertex set can be partitioned into X∪Y such that Y is an independent set, |X|<|Y| and there is an |X|-matching M such that M=xy:x∈X,y∈Y, then χρ(μt(G))=2tχρ(G). Also, we propose a special construction of a family of maximal triangle-free graph (Tn)n≥5 with typical structure and show that χρ(μt(Tn))=2tχρ(Tn).