Two graphs G and H with the same vertex set V are P4-isomorphic if every four vertices {a, b, c, d} ⊆ V induce a chordless path (denoted by P4) in G if and only if they induce a P4 in H. We call a graph split-perfect if it is P4-isomorphic to a split graph (i.e., a graph being partitionable into a clique and a stable set). This paper characterizes the new class of split-perfect graphs using the...

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2π/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered t...

We are interested in a version of graph colouring where there is a ‘co-site’ constraint value k. Given a graph G with a non-negative integral demand xv at each node v, we must assign xv positive integers (colours) to each node v such that the same integer is never assigned to adjacent nodes, and two distinct integers assigned to a single node differ by at least k. The aim is to minimise the spa...

Chen and Kanj considered the V C problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general V C to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general V C as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.0...

For a graph G = (V,E) and x : E → < satisfying ∑ e3v xe = 1 for each v ∈ V , set h(x) = ∑ e xe log(1/xe) (with log = log2). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and xe = Pr(e ∈ f), H(f) < h(x)− n 2 log e+ o(n) (where H is binary entropy). This implies a similar bound for random Hamiltonian cycles. Specializin...

Throughout the present paper, suppose FG is the group algebra of an abelian group G written multiplicatively (and possibly mixed) over a field F of positive characteristic p. As usual, FG denotes the group algebra of G over F with unit group U(FG) and normalized unit group V (FG); note that the direct decomposition U(FG) = V (FG) × F ∗ holds where F ∗ is the multiplicative group of F and thus t...

We consider the symmetry group of a Z2Z4-linear code with parameters of a 1-perfect, extended 1-perfect, or Preparata-like code. We show that, provided the code length is greater than 16, this group consists only of symmetries that preserve the Z2Z4 structure. We find the orders of the symmetry groups of the Z2Z4-linear (extended) 1-perfect codes.