Triangle-free Research Articles

Vertex disjoint copies of [formula omitted] in claw-free graphs

A complete bipartite graph with partite sets X and Y, where |X|=1 and |Y|=r, is denoted by K1,r. A graph G is said to be claw-free if G does not contain K1,3 as an induced subgraph. There are several well-known and important families of graphs that are claw-free such as line graphs and complements of triangle-free graphs. Claw-free graphs have numerous interesting properties and applications. This paper considers vertex disjoint K1,4s in claw-free graphs. Let k be an integer with k≥2 and let G be a claw-free graph with |V(G)|≥10k−9. We prove that if the minimum degree of G is at least 4, then it contains k vertex disjoint K1,4s. This result answers the question in [Jiang, Chiba, Fujita, Yan, Discrete Math. 340 (2017) 649–654].

Coloring near-quadrangulations of the cylinder and the torus

Let G be a simple connected plane graph and let C1 and C2 be cycles in G bounding distinct faces f1 and f2. For a positive integer ℓ, let r(ℓ) denote the number of integers n such that −ℓ≤n≤ℓ, n is divisible by 3, and n has the same parity as ℓ; in particular, r(4)=1. Let rf1,f2(G)=∏fr(|f|), where the product is over the faces f of G distinct from f1 and f2, and let q(G)=1+∑f:|f|≠4|f|, where the sum is over all faces f of G (of length other than four). We give an algorithm with time complexity O(rf1,f2(G)q(G)|G|) which, given a 3-coloring ψ of C1∪C2, either finds an extension of ψ to a 3-coloring of G, or correctly decides no such extension exists.The algorithm is based on a min–max theorem for a variant of integer 2-commodity flows, and consequently in the negative case produces an obstruction to the existence of the extension. As a corollary, we show that every triangle-free graph drawn in the torus with edge-width at least 21 is 3-colorable.

The chromatic number of triangle-free and broom-free graphs in terms of the number of vertices

The Garfas–Sumner conjecture asks whether for every tree T, the class of (induced) T-free graphs is $$\chi $$ -bounded. The conjecture is solved for several special trees, but it is still open in general. Motivated by the conjecture, the chromatic number of triangle-free and broom-free graphs is well studied, since a broom is one of the generalizations of a star, where a broom B(m, n) is the graph obtained from a star $$K_{1,n}$$ and an m-vertex path $$P_m$$ by identifying the center of $$K_{1,n}$$ and a leaf of $$P_m$$ . Garfas, Szemeredi and Tuza proved that for every triangle-free and B(m, n)-free graph G, $$\chi (G) \le m+n-1$$ . This upper bound has been improved by Wang and Wu to $$m+n-2$$ for $$m\ge 2, n\ge 1$$ . In this paper, we prove that any triangle-free and B(4, 2)-free graph G is 3-colorable if the number of vertices of G is at least 17. Furthermore, the above estimation is the best possible since there exists a triangle-free and B(4, 2)-free 4-chromatic graph with sixteen vertices, named the Clebsch graph.

Linearly many faults in Cayley graphs generated by transposition triangle free unicyclic graphs

We consider a kind of Cayley graphs, including the modified bubble-sort graphs MBn, the Cayley graphs generated by lollipop graphs Hn,n−1 and the other Cayley graphs generated by unicyclic triangle free graphs. We get that if we delete linearly many vertices in Cayley graphs generated by transposition unicyclic triangle free graphs, then the resulting graphs has a large connected component that comprises of nearly all residual vertices.

On Well-Dominated Graphs

A graph is well-dominated if all of its minimal dominating sets have the same cardinality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs G or H has no isolatable vertices, we prove that the direct product of G and H is well-dominated if and only if either $$G=H=K_3$$ or $$G=K_2$$ and H is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2.

Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs

Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that∑fface ofG(|f|−4)≤κ(g+t+c−1) for a fixed constant κ, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of O(g) vertices such that G−X is 3-colorable.

Upper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge Connectivity

We present upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity with respect to order and size. All bounds presented in this paper are asymptotically sharp.

On clique values identities and mantel-type theorems

‎In this paper‎, ‎we first extend the weighted handshaking‎ ‎lemma‎, ‎using a generalization of the concept of the degree of vertices to the values of graphs‎. ‎This edge-version of the weighted handshaking lemma yields an immediate generalization of the‎ ‎Mantel's classical result which asks for the maximum number of edges in triangle-free graphs‎ ‎to the class of $K_{4}$-free graphs‎. ‎Then‎, ‎by defining the concept of value‎ ‎for cliques (complete subgraphs) of higher orders‎, ‎we also‎ ‎extend the classical result of Mantel for any graph $G$‎. ‎We finally conclude our paper with a discussion‎ ‎about the possible future works‎.

Coloring triangle-free graphs with local list sizes.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Three-coloring triangle-free graphs on surfaces III. Graphs of girth five

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.

Lengths of words accepted by nondeterministic finite automata

We consider two natural problems about nondeterministic finite automata (NFA). First, given an NFA M of n states, and a length ℓ, does M accept a word of length ℓ? We show that the classic problem of triangle-free graph recognition reduces to this problem, and give an O(nω(log⁡n)1+ϵlog⁡ℓ)-time algorithm to solve it, where ω is the optimal exponent for matrix multiplication. Second, provided L(M) is finite, we consider the problem of listing the lengths of all words accepted by M. Although this problem seems like it might be significantly harder, we show that in the unary case this problem can be solved in O(nω(log⁡n)2+ϵ) time. Finally, we give a connection between NFA acceptance and the strong exponential-time hypothesis.

Acyclic Edge Coloring of IC-planar Graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G is the least number of colors such that G has an acyclic edge coloring and denoted by Χ′a(G). An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that Χ′a(G) ≤ Δ(G) + 10, if G is an IC-planar graph without adjacent triangles and Χ′a(G) ≤ Δ(G) + 8, if G is a triangle-free IC-planar graph.

Average eccentricity, minimum degree and maximum degree in graphs

Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as $$\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)$$ . We show that the average eccentricity of a connected graph of order n, minimum degree $$\delta $$ and maximum degree $$\Delta $$ does not exceed $$\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7$$ , and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.

Bipartite Induced Density in Triangle-Free Graphs

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$.
 Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, PC(2k), (that is the Cayley graph (Z22k,{e1,e2,…,e2k,J}, where the ei's are the standard basis and J=e1+e2+⋯+e2k) is a smallest graph of odd-girth 2k+1 to which every planar graph of odd-girth at least 2k+1 admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that χ′(G)=⌈χf′(G)⌉ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.Both these conjectures are believed to be true for the larger class of K5-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of K4-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth 2k+1 to admit a homomorphism from any partial t-tree of odd-girth at least 2k+1. Applying our results to the class of partial 3-trees, which is a rich subclass of K5-minor-free graphs, we prove that PC(2k) is in fact a smallest graph of odd-girth 2k+1 to which every partial 3-tree of odd-girth at least 2k+1 admits a homomorphism. We then apply this result to show that every planar (2k+1)-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is 2k+1, is (2k+1)-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.

Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in ⋃H. We give a sufficient condition for the existence of a 3-coloring ϕ of G such that for every H∈H the restriction of ϕ to H is constrained in a specified way.

Adynamic coloring of graphs

For a graph G with at least one vertex with independent neighborhood, an adynamic coloring of G is a proper vertex coloring of G such that there exists at least one vertex of degree at least 2 whose all neighbors have the same color. We explore basic properties of adynamic colorings and their relations to proper and dynamic colorings. We also establish a number of results for planar graphs; in particular, we extend the Four Color Theorem and Grötzsch’s Theorem to adynamic coloring. Finally, we prove that triangle-free graphs with maximum degree 3 are adynamically 3-colorable, which is surprisingly not true for graphs of higher maximum degrees.

A note on the ABC spectral radius of graphs

ABSTRACT The ABC matrix of a graph G, recently introduced by Estrada, is the square matrix of order whose -entry is if the ith vertex and the jth vertex of G are adjacent, and 0 otherwise, where is the degree of the ith vertex of G. The ABC spectral radius of G is the largest eigenvalue of the ABC matrix of G. In this paper, we completely characterize the (connected) graphs and (connected) triangle-free graphs which have the maximum, the minimum and second-minimum ABC spectral radius. These results answer a general question posed in [X. Chen, On ABC eigenvalues and ABC energy, Linear Algebra Appl. 544 (2018) 141–157] for (connected) graphs and (connected) triangle-free graphs.

Irreducible 4-critical triangle-free toroidal graphs

The theory of Dvořák, Král’, and Thomas (Dvořák, 2015) shows that a 4-critical triangle-free graph embedded in the torus has only a bounded number of faces of length greater than 4 and that the size of these faces is also bounded. We study the natural reduction in such embedded graphs—identification of opposite vertices in 4-faces. We give a computer-assisted argument showing that there are exactly four 4-critical triangle-free irreducible toroidal graphs in which this reduction cannot be applied without creating a triangle. Using this result, we show that every 4-critical triangle-free graph embedded in the torus has at most four 5-faces, or a 6-face and two 5-faces, or a 7-face and a 5-face, in addition to at least seven 4-faces. This result serves as a basis for the exact description of 4-critical triangle-free toroidal graphs, which we present in a followup paper.

Addendum: “The signed bad numbers in graphs”

In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.