Upper Bounds for the Automorphic Chromatic Index of a Graph

The automorphic H-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors preserved by a given subgroup H of the full automorphism group of Γ. We determine upper bounds for this index in terms of the chromatic index of Γ for some abelian 2-groups H.     

Altitude of regular graphs with girth at least five

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P. We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r?4, r-regular graphs with girth at least five have altitude at least four. Using this result we show that some snarks, including all but one of the Blanus?a type snarks, have altitude three while others, including the flower snarks, have altitude four. We construct an infinite class of 4-regular graphs with altitude four.     

On thefg-coloring of graphs

This paper introduces a new type of edge-coloring of multigraphs, called anfg-coloring, in which each color appears at each vertexv no more thanf(v) times and at each set of multiple edges joining verticesv andw no more thang(vw) times. The minimum number of colors needed tofg-color a multigraphG is called thefg-chromatic index ofG. Various upper bounds are given on thefg-chromatic index. One of them is a generalization of Vizing's bound for the ordinary chromatic index. Our proof is constructive, and immediately yields a polynomial-time algorithm tofg-color a given multigraph using colors no more than twice thefg-chromatic index.     

Decompositions and reductions of snarks

According to M. Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol. 234 (1976), pp. 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors. The problem of what “nontrivial” means is implicitly or explicitly present in most papers on snarks, and is the main motivation of the present paper. Our approach to the discussion is based on the following observation. If G is a snark with a k-edge-cut producing components G₁ and G₂, then either one of G₁ and G₂ is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G₁ and G₂ whose order does not exceed that of G. The two situations lead to a definition of a k-reduction and k-decomposition of G. Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively. The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity. The present paper is devoted to a detailed investigation of these invariants. The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6. A number of problems that have arisen in this research conclude the paper. © 1996 John Wiley & Sons, Inc.     

A Generalization of Heterochromatic Graphs and f-Chromatic Spanning Forests

In 2006, Suzuki, and Akbari and Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose f-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is f-chromatic if each color c appears on at most f(c) edges. We also present a necessary and sufficient condition for edge-colored graphs to have an f-chromatic spanning forest with exactly m components. Moreover, using this criterion, we show that a g-chromatic graph G of order n with ${|E(G)| > \binom{n-m}{2}}$ has an f-chromatic spanning forest with exactly m (1 ≤ m ≤ n ? 1) components if ${g(c) \le \frac{|E(G)|}{n-m}f(c)}$ for any color c.     

Generalized n-tuple colorings of a graph: A counterexample to a conjecture of Brigham and Dutton

In their papers (Technical Report CS-TR 50, University of Central Florida, 1980; J. Combin. Theory Ser. B32 (1982), 90–94) Brigham and Dutton introduce the notion of (n : i)-chromatic numbers of a graph, a generalization of Stahl's nth chromatic numbers (J. Combin. Theory Ser. B20, (1976), 185–203). The (n : i)-chromatic number of a graph G, denoted by χ_nⁱ(G), is the smallest integer m such that each vertex of G can be colored with a set of n colors in {1, 2,…, m} in such a way that any two adjacent vertices have exactly i colors in common. Brigham and Dutton conjecture at the end of loc cit that for all integers n and i with 0 ≤ i ≤ n ? 1, and for every graph G, χ_nⁱ⁺¹(G) ≤ χ_nⁱ(G). We prove this conjecture in some special cases and disprove it in the general case.     

On graphsG for which all large trees areG-good

LetG be a graph satisfying x(G) = k. The following problem is considered: WhichG have the property that, if n is large enough, the Ramsey numberr(G, T) has the value (k — 1)(n — 1) + 1 for all treesT onn vertices? It is shown thatG has this property if and only if for somem, G is a subgraph of bothL _k,m andM _K.m , whereL _k,m andM _k,m are two particulark-chromatic graphs. Indeed, it is shown thatr(L _k,m ,M _k,m ,T _n ) = (k — 1)(n — 1) + 1 whenn is large.     

A Class of Edge Critical 4-Chromatic Graphs

We consider several constructions of edge critical 4-chromatic graphs which can be written as the union of a bipartite graph and a matching. In particular we construct such a graph G with each of the following properties: G can be contracted to a given critical 4-chromatic graph; for each n ≥ 7, G has n vertices and three matching edges (it is also shown that such graphs must have at least \({{8n} \over 5}\) edges); G has arbitrary large girth.     

Determination of solvable algebraic groups by dense integral subgroups

Let G and G′ be triangularizable algebraic groups defined over the field Q of rational numbers, and let Γ ? G_Q and Γ′ ? G′_Q be dense subgroups of them containing integral subgroups of finite index. A study is made of the conditions under which a birational isomorphism of G and G′ follows from an abstract isomorphism of Γ and Γ′.     

Remarkable valuation of the dichromatic polynomial of planar multigraphs

Let Γ be a finite multigraph; we denote by χ(Γ, x, y) the dichromatic polynominal of Γ, as defined by W. T. Tutte in 1953. We prove that, for any planar multigraph Γ with m edges, χ(Γ, ?1, ?1) = (?1)^m · (?2)^k, where $\text{0 ≤ k ≤}\text{m}\text{2}$. Furthermore, if Γ is connected, s = k ? 1 turns out to be a pertinent invariant of the medial of Γ.     

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ? be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ?[Γ] is not a weakly factorial domain.     

On a valence problem in extremal graph theory

Let L ≠ Kinp be a p-chromatic graph and e be an edge of L such that L ? e is (p?1)-chromatic If Gⁿ is a graph of n vertices without containing L but containing K_p, then the minimum valence of Gⁿ is

$\text{?n}\text{1?}\text{1}\text{p?}\text{3}\text{2}\text{+}\text{O}\text{(1).}$

     

Note on group distance magic complete bipartite graphs

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ? from V to an Abelian group Γ of order n such that the weight $w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ? _p -distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K _m,n is a group distance magic graph if and only if n + m ? 2 (mod 4).     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

On graphs with small subgraphs of large chromatic number

Bollobás, Erdös, Simonovits, and Szemerédi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn ² edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös [2]: For each positive constantc and positive integerk there exist positive integersf _k(c) andn _o such that ifG is any graph with more thann _o vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn ² edges, thenG contains ak-chromatic subgraph with at mostf _k(c) vertices.     

The classification of complete graphsK n onf-coloring

Anf-coloring of a graphG=(V, E) is a coloring of edge setE such that each color appears at each vertexv ∈ V at mostf(v) times. The minimum number of colors needed tof-colorG is called thef-chromatic index χ′ _f (G) ofG. Any graphG hasf-chromatic index equal to Δ _f (G) or Δ _f (G) + 1 where $\Delta _f (G) = \mathop {\max }\limits_{v \in V} \left\{ {\left\lceil {\frac{{d(v)}}{{f(v)}}} \right\rceil } \right\}$ . If χ′ _f (G) = Δ _f (G), thenG is ofC _f 1; otherwiseG is ofC _f 2. In this paper, the classification problem of complete graphs onf-coloring is solved completely.     

On the existence of two non-neighboring subgraphs in a graph

Does there exist a functionf(r, n) such that each graphG with _Z (G)≧f(r, n) contains either a complete subgraph of orderr or else two non-neighboringn-chromatic subgraphs? It is known thatf(r, 2) exists and we establish the existence off(r, 3). We also give some interesting results about graphs which do not contain two independent edges.     

A survey on snarks and new results: Products,reducibility and a computer search

In this paper we survey recent results and problems of both theoretical and algorithmic character on the construction of snarks—non-trivial cubic graphs of class two, of cyclic edge-connectivity at least 4 and with girth ≥ 5. We next study the process, also considered by Cameron, Chetwynd, Watkins, Isaacs, Nedela, and Sˇkoviera, of splitting a snark into smaller snarks which compose it. This motivates an attempt to classify snarks by recognizing irreducible and prime snarks and proving that all snarks can be constructed from them. As a consequence of these splitting operations, it follows that any snark (other than the Petersen graph) of order ≤ 26 can be built as either a dot product or a square product of two smaller snarks. Using a new computer algorithm we have confirmed the computations of Brinkmann and Steffen on the classification of all snarks of order less than 30. Our results recover the well-known classification of snarks of order not exceeding 22. Finally, we prove that any snark G of order ≤ 26 is almost Hamiltonian, in the sense that G has at least one vertex v for which G \ v is Hamiltonian. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 57–86, 1998     

Minimality of horospherical flows

IfN is the nilpotent constituent of an Iwasawa decomposition of the semi-simple groupG (finite center and no compact factors), it is proved thatN acts minimally onG/Γ for every uniform lattice Γ ?G, generalizing theorems of Hedlund and L. Greenberg.