Towards finite homomorphism-homogeneousrelational structures

David Hartman 1

Department of Applied MathematicsCharles University

Prague, Czech Republic

Dragan Masulovic 2

Department of Mathematics and InformaticsUniversity of Novi Sad

Novi Sad, Serbia

Abstract

Relational structure is homomorphism-homogeneous if every local homomorphismbetween finite induced substructures can be extended to endomorphism. The clas-sification of homomorphism-homogeneous relational structures is still a challengingproblem even for a finite case. In this work finite homomorphism-homogeneousbinary relational structures having two relations that are both symmetric and ir-reflexive are classified. In addition to that, more general relational structures havingfinitely many relations of described type are considered. For those a classificationis achieved when assuming that sets of colors assigned to pairs of vertices each onerepresenting set of present edges between this pair constitute a linear partial order.

Keywords: multicolored graph, bicolored graph, homomorphism-homogeneous

1 Email: hartman@kam.mff.cuni.cz2 Email: masul@im.ns.ac.yu

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 38 (2011) 443–448

1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2011.09.072

1 Introduction

Relational structure is homomorphism-homogeneous if every local homomor-phism between finite induced substructures can be extended to endomorphismof corresponding structure [1]. This notion is extension of homogeneity whichrequires that every isomorphism between finite induced substructures can beextended to automorphism. Classification of finite homogeneous undirectedgraphs without loops was provided by Gardiner [4]. This result is also impor-tant for our case where finite homomorphism-homogeneous binary relationalstructures having two relations that are both symmetric and irreflexive areclassified and that is why we refer them as Gardiner graphs. Mentioned re-lations are often referred to as colored edges where a pair of vertices can beconnected by a red, blue or red-blue edge. Let us call such a structure abigraph and postpone more formal definition to section 3. For bigraphs thefollowing classification result has been achieved:

Theorem 1.1 A finite bigraph G = (V,E1, E2) is homomorphism-homogeneousif and only if it is one of the following:

• a disjoint union of complete graphs all having the same size and composedonly from red-blue edges,

• a connected bigraph GG = (V,E1, E2) where (V,E1) is one of the Gardinergraphs and (V,E2) is its complement,

• a bigraph that has all its connected components isomorphic to bigraphs fromthe previous case, i.e. k ·GG for some k ≥ 1.

This concept is further generalized to so called multicolored graphs thatextend bigraphs by considering more relations. This classification even for afinite case seems to be a hard task. Still partial results have been achievedwhen present colors are specifically conditioned. Roughly speaking, this con-dition requires that sets of colors representing a character of vertices’ connec-tions constitute a linear partial order, so that all edges are comparable andone can find the “maximum edge”. For a deeper description and resultingclassification see section 5.1.

2 Previous work

Symmetric mathematical structures attract attention of many mathemati-cians. Homogeneity as the strongest type of symmetric condition is quiteinteresting for classification studies.

There is a long-standing effort to classify all homogeneous relational struc-

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tures since the work of Fraisse [3]. Several of them have been classified already,e.g. the aforementioned classification of finite homogeneous undirected graphswithout loops as [4]:

(i) disjoint union of complete graphs all with the same size⋃k

i=1 Kn,

(ii) multipartite graphs Kn1,n2,...,nkwith n1 = n2 = . . . = nk,

(iii) 5-cycle C5,

(iv) line graph of bipartite graph K3,3, i.e. L(K3,3).

There are other types of homogeneous relational structures already clas-sified, such as directed graphs [2] or partially ordered sets [10]. There arealso approaches to classify homogeneous relational structures in general [7]resulting in introduction of specific contrains. However a classification of allhomogeneous relational structures still resists to be completely solved.

Recently the notion of homogeneity was modified by using homomorphisminstead of isomorphism [1]. The finite homomorphism-homogeneous undi-rected graphs were classified as [1,9] complete and null graphs. There arealso other classifications dealing with this homomorphism homogeneity, e.g.partially ordered sets [8] or finite tournaments [6]. However results for moregeneral relational structures are still missing.

3 Classified structures

General relation structures, defined as in [5], can be relaxed to so called mul-ticolored graphs. Let G = (V,E) with E = (E1, E2, . . . , Em) be a relationalstructure with domain set V and a collection of symmetric irreflexive binaryrelations E each one defined on set V . This structure is called a multicoloredgraph and relations are referred as colors. As a special case a bicolored graph,referred also as bigraph, can be defined by setting m = 2 where resulting re-lations are called red and blue. Homomorphism between multicolored graphsGA = (VA,EA) and GB = (VB,EB) is a mapping f : VA → VB such thatfor all x, y ∈ VA and all i ∈ {1, 2, . . . ,m} holds that if {x, y} ∈ Ei(GA) then{f(x), f(y)} ∈ Ei(GB). Another useful term is a decoloring of a multicoloredgraph G that denotes a graph GD = (V,ED =

⋃mi=1Ei), i.e. there is an edge

{a, b} ∈ ED if and only if there exist i such that {a, b} ∈ Ei. A graph theoret-ical property that holds for decoloring of a multicolored graph is called a weakproperty, e.g. a multicolored graph can have weak girth 5 which means thatafter decoloring the resulting graph has girth 5. On the other hand propertiesvalid for all presented relations are called (strong) properties and those valid

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for a specific relation (color) are usually denoted by pre-indexing with an indexof the given relation. For example, degi(v) denote the number of edges fromset Ei adjacent to vertex v and call this i-degree. Alternatively, color namescan be used, e.g. red degree. Other definitions are obvious generalizations ofusual terms like i-connected, i-component, etc. Specifically for i-connectivityone can call a multicolored graph (strongly) connected if it is i-connected forall i = 1, 2, . . . ,m and weakly connected if its decoloring is connected.

4 Classification of homomorpism-homogeneous bigraphs

To prove theorem 1.1 it is useful to divide the whole process into several casesaccording to presence of red-blue edges and according to weak completenessof the corresponding bigraph. At first, consider the case for which at leastone red-blue edge is present in the studied bigraph. For such a situation thefollowing proposition can be shown.

Proposition 4.1 Let G = (V,E1, E2) be a finite homomorphism-homogeneousbigraph and moreover let it contain at least one red-blue edge. Then every weakconnected component is a complete graph where each edge is red-blue and allthese components have the same number of vertices.

The core idea to prove this proposition is to show that any triple of verticescontaining a red-blue path of length 2 induces in fact a red-blue triangle.

The second case is when there is no red-blue edge and a decoloring ofthe corresponding bigraph has at least one not completely connected compo-nent. Using a slightly modified argument from the previous step and someobservations it results in following proposition.

Proposition 4.2 There is no finite homomorphism-homogeneous bigraph with-out red-blue edge whose decoloring has at least one not completely connectedcomponent.

The last case deals with bigraphs in which there is no red-blue edge andwhose decoloring has all connected components complete.

Proposition 4.3 Let G = (V,E1, E2) be a homomorphism-homogeneous bi-graph that has all its weak connected components weakly complete and withouta red-blue edge. Then all components are the same, where each one is a unionof one of the Gardiner’s graphs in one color and its complement in the othercolor.

Combining all these propositions together results in theorem 1.1 providinga classification of finite homomorphism-homogeneous bigraphs.

D. Hartman, D. Mašulovic / Electronic Notes in Discrete Mathematics 38 (2011) 443–448446

5 Multicolored graph with chain on colors

To generalize this concept let us redefine multicolored graphs in a slightlydifferent way. A multicolored graph (V,E1, . . . , Em) can be considered as apair (V, χ) where V is the set of vertices and χ : V 2 → P({1, 2, . . . ,m}) is afunction such that

(i) χ(x, x) = ∅, that is, there are no loops in the graph; and

(ii) χ(x, y) = χ(y, x) whenever x �= y, so that the graph is undirected.

The intuition is: χ(x, y) = {j : {x, y} ∈ Ej}. A homomorphism is then amapping f : (V1, χ1) → (V2, χ2) such that

χ1(x, y) ⊆ χ2(f(x), f(y))

for all x and y in V1.

In general, let L be a partially ordered set with relation � and with theleast element 0 and the greatest element 1. An L-colored graph is an orderedpair (V, χ) such that V is a nonempty set and χ : V 2 → L is a functionsatisfying the following:

• χ(x, x) = 0; and

• χ(x, y) = χ(y, x) whenever x �= y.

A homomorphism is a mapping f : (V1, χ1) → (V2, χ2) such that

χ1(x, y) � χ2(f(x), f(y))

for all x and y in V1.

For W ⊆ V , a substructure of (V, χ) induced by W is (W,χ|W ), where χ|Wdenotes the restriction of χ to W . We say that an L-colored graph G = (V, χ)is homomorphism-homogeneous if every homomorphism f : S → T betweenfinite induced subgraphs of G extends to an endomorphism of G.

5.1 Specialization to chains

Let I be a chain with the least element 0, and the greatest element 1. Letfurther G = (V, χ) be a finite I-colored graph, and let θG ⊆ V 2 be the reflexivetransitive closure of θ0G = {(x, y) ∈ V 2 : χ(x, y) 0}. Then θG is an equiva-lence relation on V whose equivalence classes will be referred to as connectedcomponents of G.

We say that a finite I-colored graph G = (V, χ) is uniform if there existsan α ∈ I \ {0} such that χ(x, x) = 0 for all x ∈ V , and χ(x, y) = α for

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all x, y ∈ V satisfying x �= y. Up to isomorphim, a finite uniform graph isuniquely determined by n = |V | and α and we denote it by U(n, α).

Using a generalization of argument for bigraphs a following theorem con-cerning I-colored graphs can be proven.

Theorem 5.1 Let G = (V, χ) be a finite I-colored graph. Then G is homo-morphism-homogeneous if and only if every vertex of G is an isolated vertex,or the following holds:

• every connected component of G is a uniform graph with at least two vertices,and

• if U(n1, α1) and U(n2, α2) are connected components of G then α1 = α2 andn1 = n2.

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