Independence in graphs with maximum degree four

Let $\text{C}$ be the class of triangle-free graphs with maximum degree four. A lower bound for the number of edges in a graph of $\text{C}$ is derived in terms of its order p and independence β. Also a characterization of certain minimum independence graphs in $\text{C}$ is provided. Let r(k) be the smallest integer such that every graph in $\text{C}$ with at least r(k) vertices has independence at least k. The values of r(k) for all k may be derived from our main theorem and $\text{4}\text{13}$ obtained as the best possible lower bound for the independence ratio $\text{β}\text{p}$ of graphs in $\text{C}$.     

k-Independence and the k-residue of a graph

Favaron, Mahéo, and Saclé proved that the residue of a simple graph G is a lower bound on its independence number α (G). For k ∈ ℕ, a vertex set X in a graph is called k-independent, if the subgraph induced by X has maximum degree less than k. We prove that a generalization of the residue, the k-residue of a graph, yields a lower bound on the k-independence number. The new bound strengthens a bound of Caro and Tuza and improves all known bounds for some graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 241–249, 1999     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

Clique-transversal sets in 4-regular claw-free graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ _c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.     

Independence Numbers of Planar Contact Graphs

Abstract. We show that for a large class of convex discs C (including strictly convex discs), there exists an ε=ε(C)>0 such that the independence number of the contact graph of any packing of n translates of C in the plane is at least (1/4 + ε)n . For C a circle, we improve the lower bound of Csizmadia to 8/31n .     

New upper bounds on harmonious colorings

We present an improved upper bound on the harmonious chromatic number of an arbitrary graph. We also consider ?fragmentable”? classes of graphs (an example is the class of planar graphs) that are, roughly speaking, graphs that can be decomposed into bounded-sized components by removing a small proportion of the vertices. We show that for such graphs of bounded degree the harmonious chromatic number is close to the lower bound (2m)^1/2, where m is the number of edges.     

Independence,odd girth,and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle‐free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle‐free graphs. For connected triangle‐free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n?m?1)/7. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96‐111, 2011     

Independence Numbers of Planar Contact Graphs

   Abstract. We show that for a large class of convex discs C (including strictly convex discs), there exists an ε=ε(C)>0 such that the independence number of the contact graph of any packing of n translates of C in the plane is at least (1/4 + ε)n . For C a circle, we improve the lower bound of Csizmadia to 8/31n .     

On lower bounds for the chromatic number in terms of vertex degree

In this paper we discuss the existence of lower bounds for the chromatic number of graphs in terms of the average degree or the coloring number of graphs. We obtain a lower bound for the chromatic number of K_1,t-free graphs in terms of the maximum degree and show that the bound is tight. For any tree T, we obtain a lower bound for the chromatic number of any K_2,t-free and T-free graph in terms of its average degree. This answers affirmatively a modified version of Problem 4.3 in [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995]. More generally, we discuss δ-bounded families of graphs and then we obtain a necessary and sufficient condition for a family of graphs to be a δ-bounded family in terms of its induced bipartite Turán number. Our last bound is in terms of forbidden induced even cycles in graphs; it extends a result in [S.E. Markossian, G.S. Gasparian, B.A. Reed, β-perfect graphs, J. Combin. Theory Ser. B 67 (1996) 1–11].     

The Crossing Number of C(mk;{1, k})

Calculating the crossing number of a given graph is, in general, an elusive problem. Garey and Johnson have proved that the problem of determining the crossing number of an arbitrary graph is NP-complete. The crossing number of a network(graph) is closely related to the minimum layout area required for the implementation of a VLSI circuit for that network. With this important application in mind, it makes most sense to analyze the the crossing number of graphs with good interconnection properties, such as the circulant graphs. In this paper we study the crossing number of the circulant graph C(mk;{1,k}) for m3, k3, give an upper bound of cr(C(mk;{1,k})), and prove that cr(C(3k;{1,k}))=k.Research supported by Chinese Natural Science Foundation     

On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

The Erd?s‐Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in the order of magnitude. Our result for the Erd?s‐Rényi graph has the following reformulation: the maximum size of a family of mutually non‐orthogonal lines in a vector space of dimension three over the finite field of order q is of order q^3/2. We also prove that every subset of vertices of size greater than q^2/2 + q^3/2 + O(q) in the Erd?s‐Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 113–127, 2007     

Lower bounds on the independence number in terms of the degrees

Wei discovered that the independence number of a graph G is at least Σ_v(1 + d(v))^?1. It is proved here that if G is a connected triangle-free graph on n ≥ 3 vertices and if G is neither an odd cycle nor an odd path, then the bound above can be increased by $\text{n}\text{Δ(Δ + 1)}$, where Δ is the maximum degree. This new bound is sharp for even cycles and for three other graphs. These results relate nicely to some algorithms for finding large independent sets. They also have a natural matrix theory interpretation. A survey of other known lower bounds on the independence number is presented.     

Graphs with the Strong Havel–Hakimi Property

The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel–Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

On the sum of all distances in a graph or digraph

The transmission of a graph or digraph G is the sum of all distances in G. Strict bounds on the transmission are collected and extended for several classes of graphs and digraphs. For example, in the class of 2-connected or 2-edge-connected graphs of order n, the maximal transmission is realized only by the cycle C_n. The independence of the transmission on the diameter or radius is shown. Remarks are also given about the NP-hardness of some related algorithmic problems.     

Maximum k-colorable subgraphs

A lower bound is established on the number of edges in a maximum k-colorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.     

On the number of cycles of length 4 in a maximal planar graph

Let p and C₄ (G) be the number of vertices and the number of 4-cycles of a maximal planar graph G, respectively. Hakimi and Schmeichel characterized those graphs G for which C₄ (G) = 1/2(p² + 3p - 22). This characterization is correct if p ≥ 9. However, for p = 7 or 8, there is exactly one other graph which violates the theorem in the sense that the upper bound of C₄ (G) is also attained.     

Extension of Turán's Theorem to the 2-Stability Number

 Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α₂(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α₂(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turán's Theorem to the q-stability number, for q>2. Received: July 21, 1999 Final version received: August 22, 2000 Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca     

Estimates of coefficients of chromatic polynomials and numbers of cliques of (c,n,m)‐graphs

This paper is mainly concerned with classes of simple graphs with exactly c connected components, n vertices and m edges, for fixed c,n,m ∈ ?. We find an optimal lower bound for the ith coefficient of the chromatic polynomial of a graph in such a class and also an optimal upper bound for the number of j‐cliques contained in such a graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 81–94, 2003     

A class of self-complementary graphs and lower bounds of some ramsey numbers

A method is described of constructing a class of self-complementary graphs, that includes a self-complementary graph, containing no K₅, with 41 vertices and a self-complementary graph, containing no K₇, with 113 vertices. The latter construction gives the improved Ramsey number lower bound r(7, 7) ≥ 114.