The homology of a locally finite graph with ends

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension 1 captures precisely the topological cycle space of graphs but works in any dimension.     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π₁(G^′) with G^′⊆G finite.     

On computing the number of linear extensions of a tree

An algorithm requiring O(n ²) arithmetic operations for computing the number of linear extensions of a poset whose covering graph is a tree is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

Lambda number of the power graph of a finite group

Journal of Algebraic Combinatorics - The power graph $$\Gamma _G$$ of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of...     

Reconstructing the number of copies of a valency-labeled finite graph in an infinite graph

Suppose that G, H are infinite graphs and there is a bijection Ψ; V(G) Ψ V(H) such that G - ξ ? H - Ψ(ξ) for every ξ ～ V(G). Let J be a finite graph and /(π) be a cardinal number for each π ? V(J). Suppose also that either /(π) is infinite for every π ? V(J) or J has a connected subgraph C such that /(π) is finite for every π ? V(C) and every vertex in V(J)/V(C) is adjacent to a vertex of C. Let (J, I, G) be the set of those subgraphs of G that are isomorphic to J under isomorphisms that map each vertex π of J to a vertex whose valency in G is /(π). We prove that the sets (J, I, G), m(J, I, H) have the same cardinality and include equal numbers of induced subgraphs of G, H respectively.     

The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits

We extend Tutte's result that in a finite 3-connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends.     

On the spanning tree packing number of a graph: a survey

The spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes.     

On the independence number of the power graph of a finite group

The power graph ${\Gamma }_G$ of a finite group $G$ is the graph whose vertex set is $G$, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ${\Gamma }_G$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_G$ if $G$ is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups $G$ whose power graphs have independence number 3 or $n?2$, where $n$ is the order of $G$.     

A relation between the comparability graph and the number of linear extensions

In this note, we prove that the comparability graph of a posetP has less edges than that of a posetQ on the same set of elements, thenP has more linear extensions thanQ. This solves a problem posed by Kelly [1].Research partially supported by the grant RP.I.09 from the Institute of Informatics, University of Warsaw.     

Remainders in extensions and finite unions of locally compact sets

We discuss the relationship between properties of spaces and their remainders in extensions from the class P _fin of all finite unions of locally compact spaces. In particular, we show that a space X ∈ P _fin iff the remainder in each (some) compactification of X is in P _fin. Then we study the class P _fin and the relationship between the remainders of a space from this class in compact extensions and give a generalization of the theorem of Henriksen-Isbell.     

Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas.     

Analysis of the physical Laplacian and the heat flow on a locally finite graph

We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence.     

Residual finiteness and hopficity of certainHNN extensions

Archiv der Mathematik -     

The semi-chromatic number of a graph

We introduce the notion of the semi-chromatic number of a graph with a nonempty number of edges. Then we prove that the difference between the semi-chromatic number and the half of the chromatic number is at most 1.