The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

Disjoint cycles in digraphs

We show that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf(k) containsk disjoint cycles. We conjecture thatf(k)=2k−1 and verify this fork=2 and we show that, for eachk≧3, the determination off(k) is a finite problem. Dedicated to Paul Erdős on his seventieth birthday     

Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

We prove that the out-distance sequence {f⁺(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f⁺(k+1)≤f⁺(k)² for k≥1, where f⁺(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertex-transitive undirected graph of infinite degree d, we have f(k)=d for all k, 1≤k<diam(G). This answers a question by L. Babai.     

On the number of homogeneous subgraphs of a graph

Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log² n) ≧G(n) ≧ exp (0.2275 log² n), the main tool is a Ramsey—Turán type theorem. We formulate a conjecture what supports Thomason’s conjecture R(k, k)^1/k = 2.     

A class of constructions for turán’s (3, 4)-problem

Letf(n) denote the minimal number of edges of a 3-uniform hypergraphG=(V, E) onn vertices such that for every quadrupleY ⊂V there existsY ⊃e ∈E. Turán conjectured thatf(3k)=k(k−1)(2k−1). We prove that if Turán’s conjecture is correct then there exist at least 2 ^k−2 non-isomorphic extremal hypergraphs on 3k vertices.     

CLASSIFICATION OF COMPLETE 5-PARTITE GRAPHS AND CHROMATICITY OF 5-PARTITE GRAPHSWITH 5n VERTICES

For a graph G,P(G,λ)denotes the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent,denoted by G-H,if P(G,λ)=p(H,λ). Let[G]= {H|H-G}. If [G]={G},then G is said to be chromatically unique. For a complete 5-partite graph G with 5n vertices, define θ(G)=(a(G,6)-2^n 1-2^n-1 5)/2n-2,where a(G,6) denotes the number of 6-independent partitions of G. In this paper, the authors show that θ(G)≥0 and determine all graphs with θ(G)= 0, 1, 2, 5/2, 7/2, 4, 17/4. By using these results the chromaticity of 5-partite graphs of the form G-S with θ(G)=0,1,2,5/2,7/2,4,17/4 is investigated,where S is a set of edges of G. Many new chromatically unique 5-partite graphs are obtained.     

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)². We conjecture that linear growth of f suffices to imply hamiltonicity.     

Contractions of infrainvariant systems of subgroups

We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G _α^* ∩G ∈ ℒ(G) for every subgroup G _α^* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

The Erdős–Pósa Property For Long Circuits

We show that, for every l, the family of circuits of length at least l satisfies the Erdős–Pósa property, with f(k)=13l(k−1)(k−2)+(2l+3)(k−1), thereby sharpening a result of C. Thomassen. We obtain as a corollary that graphs without k disjoint circuits of length l or more have tree-width O(lk²).     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Non-Separating Paths in 4-Connected Graphs

In 1975, Lovász conjectured that for any positive integer k, there exists a minimum positive integer f(k) such that, for any two vertices x, y in any f(k)-connected graph G, there is a path P from x to y in G such that G–V(P) is k-connected. A result of Tutte implies f(1) = 3. Recently, f(2) = 5 was shown by Chen et al. and, independently, by Kriesell. In this paper, we show that f(2) = 4 except for double wheels.Received October 17, 2003     

A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

Maximum set of edges no two covered by a clique

Leth(G) be the largest number of edges of the graphG. no two of which are contained in the same clique. ForG without isolated vertices it is proved that ifh(G)≦5, thenχ( )≦h(G), but ifh(G)=6 thenχ( ) can be arbitrarily large.     

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.     

Maximum degree and fractional matchings in uniform hypergraphs

Let ℋ be a family ofr-subsets of a finite setX. SetD(ℋ)= |{E:x∈E∈ℋ}|, (maximum degree). We say that ℋ is intersecting if for anyH,H′ ∈ ℋ we haveH ∩H′ ≠ 0. In this case, obviously,D(ℋ)≧|ℋ|/r. According to a well-known conjectureD(ℋ)≧|ℋ|/(r−1+1/r). We prove a slightly stronger result. Let ℋ be anr-uniform, intersecting hypergraph. Then either it is a projective plane of orderr−1, consequentlyD(ℋ)=|ℋ|/(r−1+1/r), orD(ℋ)≧|ℋ|/(r−1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.