For a graph G, let χ(G) denote its chromatic number and σ(G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of χ(G)/σ(G) over all n-vertex graphs G. A famous conjecture of Hajós from 1961 states that σ(G) ≥ χ(G) for every graph G. That is, H(n) ≤ 1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed ...

A global o ensive alliance in a graph G = (V; E) is a subset S of V such that for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. The cardinality of a minimum size global o ensive alliance in G is called the global o ensive alliance number of G. We give an upper bound on the global (strong) o ensive alliance number of a graph in terms of its degre...

An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span of f is the difference between the largest and the smallest numbers in f(V). The λ-number of G, denoted by λ(G), is the minimum span over a...

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is ‘sub-logarithmic’, then the range of such a homomorphism is super-constant. Furthermore, some exam...

In this article we consider graph partitions with prescribed patterns. Specifically a 2-colouring of the vertex set such that one colour induces an independent and other complete bipartite graph. We determine list obstructions (minimal forbidden induced subgraphs) for chordal graphs. This result completes knowledge on minimal 3 × matrix partition problems

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n=h vertex disjoint copies of H. The fractional arboricity of H is a(H) = maxf jE 0 j jV 0 j?1 g, where the maximum is taken over all subgraphs (V 0 ; E 0) of H with jV 0 j > 1. Let (H) denote the minimum degree of a vertex of H. It is shown that if (H) < a(H) then n ?1...

The problem of finding a minimum weight k-vertex connected spanning subgraph is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive polynomial time approximation ...

Let A ∈ Cn be an exceptional extremal copositive n × n matrix with positive diagonal. A zero u of A is a non-zero nonnegative vector such that uTAu = 0. The support of a zero u is the index set of the positive elements of u. A zero u is minimal if there is no other zero v such that supp v ⊂ suppu strictly. Let G be the graph on n vertices which has an edge (i, j) if and only if A has a zero wit...

For a graph G, let χ(G) denote its chromatic number and σ(G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of χ(G)/σ(G) over all n-vertex graphs G. A famous conjecture of Hajós from 1961 states that σ(G) ≥ χ(G) for every graph G. That is, H(n) ≤ 1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed ...

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is a(H) = max{ |E ′| |V ′|−1}, where the maximum is taken over all subgraphs (V ′, E′) of H with |V ′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H) then n−1/a(H)...