K4-free Graphs Research Articles

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.

Improved bounds on the chromatic number of (P5, flag)-free graphs

Given a positive integer t, let Pt and Kt respectively denote the chordless path and the complete graph on t vertices. For a graph G, let χ(G) and ω(G) respectively denote the chromatic number and clique number of G. It is known that every (P5,K4)-free graph G satisfies χ(G)≤5, and the bound is tight. A flag is the graph obtained from a K4 by attaching a pendent vertex. Clearly, the class of flag-free graphs generalizes the class of K4-free graphs. In this paper, we show the following:•Every (P5,flag,K5)-free graph G that contains a K4 satisfies χ(G)≤8.•Every (P5,flag,K6)-free graph G satisfies χ(G)≤8.•Every (P5,flag,K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the given bounds are tight. Further, we show that every (P5, flag)-free graph G with ω(G)≥4 satisfies χ(G)≤max⁡{8,2ω(G)−3}, and the bound is tight for ω(G)∈{4,5,6}. We note that our bound is an improvement over that given in Dong et al. [3,4].

Edge clique partition in [formula omitted]-graphs

An edge clique partition of G is a set of edge-disjoint cliques of G which together contain every edge of G. In this work we investigate the computational complexity of determining the minimum size of an edge clique partition of G, denoted by cp(G). This problem is NP-hard for split graphs and K4-free graphs. We establish a complete complexity classification of this problem on the hierarchy of (k,ℓ)-graphs, graphs that can be partitioned into k independent sets and ℓ complete sets. In addition, we determine in polynomial time the value of cp(G) for interesting subclasses of (k,ℓ)-graphs: split-indifference graphs, partial 2-trees, and triangular grids.

On graphic arrangement groups

A finite simple graph Γ determines a quotient PΓ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a K4-free graph Γ, a product of deletion maps is injective, embedding PΓ in a product of free groups. Then PΓ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show PΓ is of homological finiteness type Fm−1, but not Fm, where m is the number of copies of K3 in Γ, except in trivial cases. The embedding result is extended to graphs whose 4-cliques share at most one edge, giving an injection of PΓ into the product of pure braid groups corresponding to maximal cliques of Γ. We give examples showing that this map may inject in more general circumstances. We define the graphic braid group BΓ as a natural extension of PΓ by the automorphism group of Γ, and extend our homological finiteness result to these groups.

K4-free character graphs with seven vertices

For a finite group G, let Δ(G) denote the character graph built on the set of degrees of the irreducible complex characters of G. In this article, we determine the structure of all finite groups G with K4-free character graph Δ(G) having seven vertices. We also obtain a classification of all K4-free graphs with seven vertices which can occur as character graphs of some finite groups.

Clique roots of K4-free chordal graphs

The clique polynomial C ( G , x ) of a finite, simple and undirected graph G = ( V , E ) is defined as the ordinary generating function of the number of complete subgraphs of G . A real root of C ( G , x ) is called a clique root of the graph G . Hajiabolhasan and Mehrabadi showed that every simple graph G has at least a clique root in the interval [ − 1, 0) . Moreover, they showed that the class of triangle-free graphs has only clique roots. In this paper, we extend their result by showing that the class of K 4 -free chordal graphs has also only clique roots. In particular, we show that this class has always a clique root − 1 . We conclude our paper with some interesting open questions and conjectures.

On the number of edge-disjoint triangles in K4-free graphs

We prove the quarter of a century old conjecture of Erdős that every K4-free graph with n vertices and ⌊n2/4⌋+m edges contains m pairwise edge disjoint triangles.

Bounded clique cover of some sparse graphs

We show that f(x)=⌊32x⌋ is a θ-bounding function for the class of subcubic graphs and that it is best possible. This generalizes a result by Henning et al. (2012), who showed that θ(G)≤32α(G) for any subcubic triangle-free graph G. Moreover, we provide a θ-bounding function for the class of K4-free graphs with maximum degree at most 4. Finally, we study the problem Clique Cover for subclasses of planar graphs and graphs with bounded maximum degree: in particular, answering a question of Cerioli et al. (2008), we show it admits a PTAS for planar graphs.

On the Number of Edge-Disjoint Triangles in K4-Free Graphs

We prove the quarter of a century old conjecture that every K4-free graph with n vertices and [n2/4]+m edges contains m pairwise edge-disjoint triangles.

On the number of [formula omitted]-saturating edges

Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any n-vertex K4-free graph G with ⌊n2/4⌋+1 edges, one can find at least (1+o(1))n216K4-saturating edges. We construct a graph with only 2n233K4-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges.

A stability theorem on fractional covering of triangles by edges

Let ν(G) denote the maximum number of edge-disjoint triangles in a graph G and τ∗(G) denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that τ∗(G)≤2ν(G) for every graph G. This is sharp, since for the complete graph K4 we have ν(K4)=1 and τ∗(K4)=2. We refine this result by showing that if a graph G has τ∗(G)≥2ν(G)−x, then G contains ν(G)−⌊10x⌋ edge-disjoint K4-subgraphs plus an additional ⌊10x⌋ edge-disjoint triangles. Note that just these K4’s and triangles witness that τ∗(G)≥2ν(G)−⌊10x⌋. Our proof also yields that τ∗(G)≤1.8ν(G) for each K4-free graph G. In contrast, we show that for each ϵ>0, there exists a K4-free graph Gϵ such that τ(Gϵ)>(2−ϵ)ν(Gϵ).

Packing and Covering Triangles in K 4-free Planar Graphs

We show that every K 4-free planar graph with at most ν edge-disjoint triangles contains a set of at most $${\frac32\nu}$$ edges whose removal makes the graph triangle-free. Moreover, equality is attained only when G is the edge-disjoint union of 5-wheels plus possibly some edges that are not in triangles. We also show that the same statement is true if instead of planar graphs we consider the class of graphs in which each edge belongs to at most two triangles. In contrast, it is known that for any c < 2 there are K 4-free graphs with at most ν edge-disjoint triangles that need more than cν edges to cover all triangles.

K4-free graphs with no odd hole: Even pairs and the circular chromatic number

An odd hole in a graph is an induced cycle of odd length at least five. In this article we show that every imperfect K4-free graph with no odd hole either is one of two basic graphs, or has an even pair or a clique cutset. We use this result to show that every K4-free graph with no odd hole has circular chromatic number strictly smaller than 4. We also exhibit a sequence {Hn} of such graphs with limn→∞χc(Hn)=4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 303–322, 2010

Formula omitted]-partition of some classes of graphs

We consider the problem of partitioning the vertex-set of a graph into four non-empty sets A,B,C,D such that every vertex of A is adjacent to every vertex of B and every vertex of C is adjacent to every vertex of D. The complexity of deciding if a graph admits such a partition is unknown. We show that it can be solved in polynomial time for several classes of graphs: K4-free graphs, diamond-free graphs, planar graphs, graphs with bounded treewidth, claw-free graphs, (C5,P5)-free graphs and graphs with few P4’s.

Clique-coloring some classes of odd-hole-free graphs

We consider the problem of clique-coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd-hole-free graph that is not 3-clique-colorable, the existence of a constant C such that any perfect graph is C-clique-colorable is an open problem. In this paper we solve this problem for some subclasses of odd-hole-free graphs: those that are diamond-free and those that are bull-free. We also prove the NP-completeness of 2-clique-coloring K4-free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006

On Clique-inverse graphs of Kp-free graphs

Abstract In this paper, for p ≥ 4 , we prove that the family of forbidden induced subgraphs for a graph G such that the clique graph of G is Kp-free is finite, providing a proof of a conjecture in [F. Protti and J. Szwarcfiter. Clique-Inverse Graphs of K3-free and K4-free Graphs, Journal of Graph Theory (2000) 257–272]. We also give an upper bound for the rank of Sperner conformal hypergraphs that contain p pairwise intersecting edges and are minimal with respect to all these properties.

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique-inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique-inverse graphs of K3-free and K4-free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000

Clique-inverse graphs ofK3-free andK4-free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique-inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique-inverse graphs of K3-free and K4-free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000

The maximum number of triangles in a K4-free graph

Let v, e and t denote the number of vertices, edges and triangles, respectively, of a K 4-free graph. Fisher (1988) proved that t ⩽ ( e/3) 3/2, independently of v. His bound is attained when e = 3 k 2 for some integer k, but not in general. We find here, for any given value of e, the maximum possible value of t. Again, the maximum does not depend on v. We also show that certain ‘gaps’ occur in the set of triples ( v, e, t) realizable by K 4-free graphs.

Degree multiplicities and independent sets in K4-free graphs

Erdős et al. [6] asked for what values of k there is a sequence of graphs ( G n ) n ∞ = 1 such that G n has n vertices and independence number o( n), contains no K 4, and no k + 1 vertices have the same degree. In [6] it is proved that such a k has to be at least 4, but the problem whether any such k exists is left open. By making use of some graphs constructed by Bollobás and Erdős [3], we shall prove that for k = 5 (and so for k ⩾ 5) there is such a sequence ( G n ) n ∞ = 1.