Induced trees in graphs of large chromatic number

Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T) such that every graph G with χ(G) > f(k, t) contains either K_k or an induced copy of T. We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k,T) such that every graph G with χ(G) > g(k,t) contains either K_k or an induced copy of a subdivision of T. © 1997 John Wiley & Sons, Inc.     

Subdivisions of graphs with large minimum degree

We prove a theorem on path systems implying the conjecture of Bollobás that there exists a function f(k, m) (where k and m are natural numbers) satisfying the following: For each graph G of minimum degree at least f(k, m) there exists a graph H of minimum degree at least k such that G contains the graph obtained from H by subdividing each edge m times.     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

Packing chromatic number versus chromatic and clique number

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in [k]\), where each \(V_i\) is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that \(\omega (G) = a\), \(\chi (G) = b\), and \(\chi _{\rho }(G) = c\). If so, we say that (a, b, c) is realizable. It is proved that \(b=c\ge 3\) implies \(a=b\), and that triples \((2,k,k+1)\) and \((2,k,k+2)\) are not realizable as soon as \(k\ge 4\). Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on \(\chi _{\rho }(G)\) in terms of \(\Delta (G)\) and \(\alpha (G)\) is also proved.     

An upper bound for the total chromatic number

The total chromatic number,(G), of a graphG, is defined to be the minimum number of colours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that , where(G) is the vertex chromatic number and(G) is the edge chromatic number of the graph.Partially supported by ORS grant ORS/84120     

On the complexity of the circular chromatic number

Circular chromatic number, χ_c is a natural generalization of chromatic number. It is known that it is NP ‐hard to determine whether or not an arbitrary graph G satisfies χ(G)=χ_c(G). In this paper we prove that this problem is NP ‐hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G) = n, it is NP ‐complete to verify if . © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 226–230, 2004     

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

If G is a connected graph of order n ⩾ 1, then by a hamiltonian coloring of G we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| ⩾ n − 1 − D _G (x, y) (where D _G (x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ V (G). Let G be a connected graph. By the hamiltonian chromatic number of G we mean

On the chromatic number of the general Kneser-graph

Given integers k, n, 2 < k < n, let us define a graph with vertex set V = {F ?{1, 2, …, n}: ∩F = k}, and (F, F') is an edge if |F ∩ F′| ≤ 1. We show that for n > n₀(k) the chromatic number of this graph is (k - 1)() + rs, where n = (k - 1)s + r, 0 ≤ r < k - 1.     

Finite fields and the 1‐chromatic number of orientable surfaces

The 1‐chromatic number χ₁(S_p) of the orientable surface S_p of genus p is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. We show that if there exists a finite field of order 4m+1, m≥3, then 8m+2≤χ₁(S)≤8m+3, where 8m+3 is Ringel's upper bound on χ₁(S). © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 179–184, 2010     

Factorizing the complete graph into factors with large star number

The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that $\text{c}{}_1{}^{\mathrm{n}}\text{k ≤ f(n, k) < c}{}_2{}^{\mathrm{n}}\text{k}$.     

Nearly bipartite graphs with large chromatic number

P. Erdős and A. Hajnal asked the following question. Does there exist a constant ε>0 with the following property: If every subgraphH of a graphG can be made bipartite by the omission of at most ε|H| edges where |H| denotes the number of vertices ofH thenx(H) ≦ 3. The aim of this note is to give a negative answer to this question and consider the analogous problem for hypergraphs. The first was done also by L. Lovász who used a different construction.     

The chromatic number of oriented graphs

We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) defined as the minimum order of an oriented graph H such that G admits a homomorphism to H. We study the chromatic number of oriented k-trees and of oriented graphs with bounded degree. We show that there exist oriented k-trees with chromatic number at least 2^k+1 - 1 and that every oriented k-tree has chromatic number at most (k + 1) × 2^k. For 2-trees and 3-trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k - 1) × 2^2k-2. For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 191–205, 1997     

On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

On the Ramsey number of trees versus graphs with large clique number

Chvátal established that r(T_m, K_n) = (m – 1)(n – 1) + 1, where T_m is an arbitrary tree of order m and K_n is the complete graph of order n. This result was extended by Chartrand, Gould, and Polimeni who showed K_n could be replaced by a graph with clique number n and order n + 1 provided n ≧ 3 and m ≧ 3. We further extend these results to show that K_n can be replaced by any graph on n + 2 vertices with clique number n, provided n ≧ 5 and m ≧ 4. We then show that further extensions, in particular to graphs on n + 3 vertices with clique number n are impossible. We also investigate the Ramsey number of trees versus complete graphs minus sets of independent edges. We show that r(T_m, K_n –tK₂) = (m – 1)(n – t – 1) + 1 for m ≧ 3, n ≧ 6, where T_m is any tree of order m except the star, and for each t, O ≦ t ≦ [(n – 2)/2].     

Simultaneous approximation in number fields

Summary We present here the solution of a problem of J.-J. Sansuc together with a natural generalization of it. This problem of Sansuc is, given a number fieldk, to find the smallest positive integerm for which there exists a finitely generated subgroup of rankm ofk ^x having a dense image in (R _Q k)^x under the canonical embedding. This integer is the number of archimedean places ofk plus one.Oblatum 28-I-1992This work was partially supported by NSERC and FCAR grants     

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

Generalized degree conditions for graphs with bounded independence number

We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K_1,m-free grphs we obtain generalizations of known results. In particular we show: Theorem. Let r ≥ 1 and m ≥ 3 be integers. Then for each nonnegative function f(r, m) there exists a constant C = C(r, m, f(r, m)) such that if G is a graph of order n (n ≥ r, n > m) with δ_r(G) ≥ (n/3) + C and β (G) ≥ f(r, m), then (a) G is traceable if δ(G) ≥ r and G is connected; (b) G is hamiltonian if δ(G) ≥ r + 1 and G is 2-connected; (c) G is hamiltonian-connected if δ(G) ≥ r + 2 and G is 3-connected. © 1995 John Wiley & Sons, Inc.     

Adaptable chromatic number of graph products

A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1,2,…,k admits an adapted vertex-colouring of G by the same colours 1,2,…,k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. While the ordinary chromatic number of all (categorical) powers G^k of G remains the same as that of G, the adaptable chromatic number of G^k may increase with k. We conjecture that for all sufficiently large k the adaptable chromatic number of G^k equals the chromatic number of G. When G is complete, we prove this conjecture with k≥4, and offer additional evidence suggesting it may hold with k≥2. We also discuss other products and propose several open problems.     

The chromatic Ramsey number of odd wheels

We prove that the chromatic Ramsey number of every odd wheel W_{2k+ 1}, k?2 is 14. That is, for every odd wheel W_{2k+ 1}, there exists a 14‐chromatic graph F such that when the edges of F are two‐coloured, there is a monochromatic copy of W_{2k+ 1} in F, and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski graphs could help to prove the Burr–Erd?s–Lovász conjecture on the minimum possible chromatic Ramsey number of an n‐chromatic graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:198‐205, 2012     

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory