Covering line graphs with equivalence relations

An equivalence graph is a disjoint union of cliques, and the equivalence number of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds: . This disproves a recent conjecture that is at most three for triangle-free G; indeed it can be arbitrarily large.To bound we bound the closely related invariant σ(G), which is the minimum number of orientations of G such that for any two edges e,f incident to some vertex v, both e and f are oriented out of v in some orientation. When G is triangle-free, . We prove that even when G is triangle-free, it is NP-complete to decide whether or not σ(G)≤3.     

On mean distance and girth

We bound the mean distance in a connected graph which is not a tree in terms of its order n and its girth g. On one hand, we show that the mean distance is at most if g is even and at most if g is odd. On the other hand, we prove that the mean distance is at least unless G is an odd cycle. This resolves two conjectures of AutoGraphiX.     

A note on the hardness of Skolem-type sequences

The purpose of this note is to give upper bounds (assuming different from ) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi-Skolem sequences and generalized Skolem sequences are strongly -complete. These results are significant strengthenings and simplifications of the recent -completeness result for generalized multi-Skolem sequences.     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

Forwarding and optical indices of a graph

Motivated by wavelength-assignment problems for all-to-all traffic in optical networks, we study graph parameters related to sets of paths connecting all pairs of vertices. We consider sets of both undirected and directed paths, under minimisation criteria known as edge congestion and wavelength count; this gives rise to four parameters of a graph G: its edge forwarding index π(G), arc forwarding index , undirected optical index , and directed optical index .In the paper we address two long-standing open problems: whether the equality holds for all graphs, and whether indices π(G) and are hard to compute. For the first problem, we give an example of a family of planar graphs {G_k} such that . For the second problem, we show that determining either π(G) or is NP-hard.     

Two classes of edge domination in graphs

Let (, resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

On strong unimodality of multivariate discrete distributions

A discrete function f defined on Zⁿ is said to be logconcave if for , , . A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189-216] a discrete function is called strongly unimodal if there exists a convex function such that  if . In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.     

The connectivity of a graph and its complement

Let G be a graph with minimum degree δ(G), edge-connectivity λ(G), vertex-connectivity κ(G), and let be the complement of G.In this article we prove that either λ(G)=δ(G) or . In addition, we present the Nordhaus-Gaddum type result . A family of examples will show that this inequality is best possible.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

Periodicity of Adams operations on the Green ring of a finite group

The Adams operations and on the Green ring of a group G over a field K arise from the study of the exterior powers and symmetric powers of KG-modules. When G is finite and K has prime characteristic p we show that and are periodic in n if and only if the Sylow p-subgroups of G are cyclic. In the case where G is a cyclic p-group we find the minimum periods and use recent work of Symonds to express in terms of .     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).