Long paths in sparse random graphs

We consider random graphs withn labelled vertices in which edges are chosen independently and with probabilityc/n. We prove that almost every random graph of this kind contains a path of length ≧(1 −α(c))n where α(c) is an exponentially decreasing function ofc. Dedicated to Tibor Gallai on his seventieth birthday     

Lower bound of the hadwiger number of graphs by their average degree

The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

On-line coloring of perfect graphs

Lovász, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-linek-colorable graph onn vertices withO(nlog^(2k–3) n/log^(2k–4) n) colors. Vishwanathan showed that at least (log ^k–1 n/k ^k ) colors are needed. While these remain the best known bounds, they give a distressingly weak approximation of the number of colors required. In this article we study the case of perfect graphs. We prove that there exists an on-line algorithm which will color any on-linek-colorable perfect graph onn vertices withn ^10k/loglogn colors and that Vishwanathan's techniques can be slightly modified to show that his lower bound also holds for perfect graphs. This suggests that Vishwanathan's lower bound is far from tight in the general case.Research partially supported by Office of Naval Research grant N00014-90-J-1206.     

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.     

Independent sets ink-chromatic graphs

A graph is said to have propertyP _k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP _k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP _k orP _k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP _k or is a circuit of length 7. Dedicated to Paul Erdős on his seventieth birthday     

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.     

On the minimal number of edges in color-critical graphs

A graphG isk-critical if it has chromatic numberk, but every proper subgraph of it is (k?1)-colorable. This paper is devoted to investigating the following question: for givenk andn, what is the minimal number of edges in ak-critical graph onn vertices, with possibly some additional restrictions imposed? Our main result is that for everyk≥4 andn>k this number is at least $\left( {\frac{{k - 1}}{2} + \frac{{k - 3}}{{2(k^2 - 2k - 1)}}} \right)n$ , thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges ink-critical graphs and provide some constructions of sparsek-critical graphs. A few applications of the results to Ramsey-type problems and problems about random graphs are described.     

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.     

An approximate Dirac-type theorem for k-uniform hypergraphs

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hypergraphs: For every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges. Research supported by NSF grant DMS-0300529. Research supported by KBN grant 2P03A 015 23 and N201036 32/2546. Part of research performed at Emory University, Atlanta. Research supported by NSF grant DMS-0100784.     

A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free

A graph G is diameter 2-critical if its diameter is 2, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements are claw-free.     

Contractible edges in triangle-free graphs

An edge of a graph is calledk-contractible if the contraction of the edge results in ak-connected graph. Thomassen [5] proved that everyk-connected graph of girth at least four has ak-contractible edge. In this paper, we study the distribution ofk-contractible edges in triangle-free graphs and show the following: Whenk≧2, everyk-connected graph of girth at least four and ordern≧3k, hasn+(3/2)k ²-3k or morek-contractible edges.     

Diperfect graphs

Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ₁, μ₂, ..., μ_k such tht |μ_i ∩S|=1 for alli. Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S ₁,S ₂, ...,S _k) and a path μ such that |μ ∩S _i|=1 for alli. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture. Dedicated to Tibor Gallai on his seventieth birthday     

Large faces in 4-critical planar graphs with minimum degree 4

We prove that the size of the largest face of a 4-critical planar graph with 4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows .All three authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada.     

Regular subgraphs of dense graphs

Every graph onn vertices, with at leastc _k n logn edges contains ak-regular subgraph. This answers a question of Erdős and Sauer.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Disjoint shortest paths in graphs

It is an interesting problem that how much connectivity ensures the existence ofn disjoint paths joining givenn pairs of vertices, but to get a sharp bound seems to be very difficult. In this paper, we study how muchgeodetic connectivity ensures the existence ofn disjointgeodesics joining givenn pairs of vertices, where a graph is calledk-geodetically connected if the removal of anyk−1 vertices does not change the distance between any remaining vertices.     

Equiarboreal graphs

A graphX is said to beequiarboreal if the number of spanning trees containing a specified edge inX is independent of the choice of edge. We prove that any graph which is a colour class in an association scheme (and thus any distance regular graph) is equiarboreal. We note that a connected equiarboreal graph withM edges andn vertices has edge-connectivity at leastM/(n−1).     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.