Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

A good submatrix is hard to find

Given a matrix, it is NP-hard to find a ‘large’ column, row, or arbitraty submatrix that satisfies property π, where π is nontrivial, holds for permutation matrices, and is hereditary on submatrices. Such properties include totally unimodular, transformable to a network matrix, permutable to consecutive ones, and many others. Similar results hold for properties such as positive definite, of bandwidth w, and symmetric.     

What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

A Note on the Universal Factorization Property of Groups

We give criteria when a full subcategory D of the category of groups has C-universal factorization property (C-UFP) or C-strong universal factorization property(C-SUFP) for a certain category of groups C.As a byproduct,we give affirmative answers to three unsettled questions in[S.W.Kim,J.B.Lee,Universal factorization property of certain polycyclic groups,J.Pure Appl.Algebra 204 (2006) 555-567].     

Diamond-free circle graphs are Helly circle

The diamond is the graph obtained from K₄ by deleting an edge. Circle graphs are the intersection graphs of chords in a circle. Such a circle model has the Helly property if every three pairwise intersecting chords intersect in a single point, and a graph is Helly circle if it has a circle model with the Helly property. We show that the Helly circle graphs are the diamond-free circle graphs, as conjectured by Durán. This characterization gives an efficient recognition algorithm for Helly circle graphs.