Cycles through specified vertices of a graph

We prove that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG. Moreover, whenk≧3, each cycle ofG can be expressed as the sum of an odd number of cycles throughS. On the other hand, ifS is a set ofk vertices, these conclusions do not necessarily hold, and we characterize the exceptional cases. As corollaries, we establish the existence of odd and even cycles through specified vertices and deduce the existence of long odd and even cycles in graphs of high connectivity.     

Precoloring extension for K4‐minor‐free graphs

Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K₄‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009     

Legal coloring of graphs

The following computational problem was initiated by Manber and Tompa (22nd FOCS Conference, 1981): Given a graphG=(V, E) and a real functionf:V→R which is a proposed vertex coloring. Decide whetherf is a proper vertex coloring ofG. The elementary steps are taken to be linear comparisons. Lower bounds on the complexity of this problem are derived using the chromatic polynomial ofG. It is shown how geometric parameters of a space partition associated withG influence the complexity of this problem. Existing methods for analyzing such space partitions are suggested as a powerful tool for establishing lower bounds for a variety of computational problems.     

Bi-complement Reducible Graphs

We introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG = (W ∪ B, E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphofGis the graph having the same vertex setW ∪ BasG, while its edge set is equal toW × B − E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs.     

Embedding finite graphs into graphs colored with infinitely many colors

We prove that for any cardinalτ and for any finite graphH there is a graphG such that for any coloring of the pairs of vertices ofG withτ colors there is always a copy ofH which is an induced subgraph ofG so that both the edges of the copy and the edges of the complement of the copy are monochromatic. Research supported by Hungarian National Science Foundation OTKA grant 1805.     

On the graph of large distances

For a setS of points in the plane, letd ₁>d ₂>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd _k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|constk ². IfS consists of the vertices of a convex polygon and |S|constk ², then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k – 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.     

Matching behaviour is asymptotically normal

Ak-matching in a graphG is a set ofk edges, no two of which have a vertex in common. The number of these inG is writtenp(G, k). Using an idea due to L. H. Harper, we establish a condition under which these numbers are approximately normally distributed. We show that our condition is satisfied ifn=|V(G)| is large compared to the maximum degree Δ of a vertex inG(i.e. Δ=o(n)) orG is a large complete graph. One corollary of these results is that the number of points fixed by a randomly chosen involution in the symmetric groupS is asymptotically normally distributed.     

Some remarks on (k−1)-critical subgraphs ofk-critical graphs

A graphG is said to bek-critical if it has chromatic numberk, but every proper subgraph ofG has a (k–1)-coloring. Gallai asked whether every largek-critical graph contains many (k–1)-critical subgraphs. We provide some information concerning this question and some related questions.     

Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs

We present in this article an evolutionary procedure for solving general optimization problems. The procedure combines efficiently the mechanism of a simple descent method and of genetic algorithms. In order to explore the solution space properly, periods of optimization are interspersed with phases of interaction and diversification. An adaptation of this search principle to coloring problems in graphs is discussed. More precisely, given a graphG, we are interested in finding the largest induced subgraph ofG that can be colored with a fixed numberq of colors. Whenq is larger or equal to the chromatic number ofG, then the problem amounts to finding an ordinary coloring of the vertices ofG.     

On a connection between the existence ofk-trees and the toughness of a graph

A graphG has toughnesst(G) ift(G) is the largest numbert such that for any subsetS of the vertices ofG, the number of vertices inS is at leastt times the number of components ofG on deletion of the vertices inS, provided that there is then more than one component. Ak-tree of a connected graph is a spanning tree with maximum degree at mostk. We show here that if , withk 3, thenG has ak-tree. The notion of ak-tree generalizes the casek = 2 of a hamiltonian path, so that this result, as we discuss, may be of some interest in connection with Chvátal's conjecture that, for somet, every graph with toughness at leastt is hamiltonian.     

On the<Emphasis Type="Italic">k</Emphasis>-systems of a simple polytope

Ak-system of the graphG _P of a simple polytopeP is a set of induced subgraphs ofG _P that shares certain properties with the set of subgraphs induced by thek-faces ofP. This new concept leads to polynomial-size certificates in terms ofG _P for both the set of vertex sets of facets and for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.     

Edge covering coloring of nearly bipartite graphs

LetG be a simple graph with vertex setV(G) and edge setE(G). A subsetS ofE(G) is called an edge cover ofG if the subgraph induced byS is a spanning subgraph ofG. The maximum number of edge covers which form a partition ofE(G) is called edge covering chromatic number ofG, denoted by χ′_c(G). It known that for any graphG with minimum degreeδ,δ -1 ≤χ′_c(G) ≤δ. If χ′_c(G) =δ, thenG is called a graph of CI class, otherwiseG is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.     

Pebbling numbers of some graphs

Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.     

Ramsey numbers for local colorings

The concept of a localk-coloring of a graphG is introduced and the corresponding localk-Ramsey numberr _loc ^k (G) is considered. A localk-coloring ofG is a coloring of its edges in such a way that the edges incident to any vertex ofG are colored with at mostk colors. The numberr _loc ^k (G) is the minimumm for whichK _m contains a monochromatic copy ofG for every localk-coloring ofK _m . The numberr _loc ^k (G) is a natural generalization of the usual Ramsey numberr ^k (G) defined for usualk-colorings. The results reflect the relationship betweenr ^k (G) andr _loc ^k (G) for certain classes of graphs.This research was done while under an IREX grant.     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

A generalization of menger's theorem for certain block—cactus graphs

A graphG is called a block—cactus graph if each block ofG is complete or a cycle. In this paper, we shall show that a block—cactus graphG has the property that the cardinality of a smallest set separating any vertex setJ ofG is the maximum number of internally disjoint paths between the vertices ofJ if and only if every block ofG contains at most two cut-vertices. This result extends two theorems of Sampathkumar [4] and [5].     

Equitable list coloring and treewidth

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most ?|V(G)|/k? vertices. A graph is equitably k ‐choosable if such a coloring exists whenever the lists all have size k. Kostochka, Pelsmajer, and West introduced this notion and conjectured that G is equitably k‐choosable for k>Δ(G). We prove this for graphs of treewidth w≤5 if also k≥3w?1. We also show that if G has treewidth w≥5, then G is equitably k‐choosable for k≥max{Δ(G)+w?4, 3w?1}. As a corollary, if G is chordal, then G is equitably k‐choosable for k≥3Δ(G)?4 when Δ(G)>2. © 2009 Wiley Periodicals, Inc. J Graph Theory     

Linear upper bounds for local Ramsey numbers

A coloring of the edges of a graph is called alocal k-coloring if every vertex is incident to edges of at mostk distinct colors. For a given graphG, thelocal Ramsey number, r _loc ^k (G), is the smallest integern such that any localk-coloring ofK _n , (the complete graph onn vertices), contains a monochromatic copy ofG. The following conjecture of Gyárfás et al. is proved here: for each positive integerk there exists a constantc = c(k) such thatr _loc ^k (G) cr ^k (G), for every connected grraphG (wherer ^k (G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.On leave from the Institute of Mathematics, Technical University of Warsaw, Poland.While on leave at University of Louisville, Fall 1985.     

Seidel-equivalence inLB 3(6) graphs

It is proved that the Seidel-equivalent of aLB ₃(6) graphG with respect to a partition (W ₁,W ₂) of its vertex set is a pseudo-LB ₃(6) graph if and only ifW ₁ consists of six vertices lying on a line ofG. Research supported by NSF Grant GP-20537. At present at Department of Mathematics, University of Bombay, India.