Disconnected Colors in Generalized Gallai‐Colorings

Gallai‐colorings of complete graphs—edge colorings such that no triangle is colored with three distinct colors—occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper), information theory and the theory of perfect graphs. A basic property of Gallai‐colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.     

Acyclic improper choosability of graphs

We consider improper colorings (sometimes called generalized, defective or relaxed colorings) in which every color class has a bounded degree. We propose a natural extension of improper colorings: acyclic improper choosability. We prove that subcubic graphs are acyclically (3, 1)*-choosable (i.e. they are acyclically 3-choosable with color classes of maximum degree one). Using a linear time algorithm, we also prove that outerplanar graphs are acyclically (2, 5)*-choosable (i.e. they are acyclically 2-choosable with color classes of maximum degree five). Both results are optimal. We finally prove that acyclic choosability and acyclic improper choosability of planar graphs are equivalent notions.     

Rainbow matchings in Dirac bipartite graphs

We show the existence of rainbow perfect matchings in μn‐bounded edge colorings of Dirac bipartite graphs, for a sufficiently small μ > 0. As an application of our results, we obtain several results on the existence of rainbow k‐factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.     

Perfect 2-colorings of a hypercube

A coloring of the vertices of a graph is called perfect if the multiset of colors of all neighbors of a vertex depends only on its own color. We study the possible parameters of perfect 2-colorings of the n-dimensional cube. Some necessary conditions are obtained for existence of such colorings. A new recursive construction of such colorings is found, which produces colorings for all known and infinitely many new parameter sets.     

On Perfect 2‐Colorings of Johnson Graphs J(v, 3)

We study the perfect 2‐colorings (also known as the equitable partitions into two parts or the completely regular codes with covering radius 1) of the Johnson graphs . In particular, we classify all the realizable quotient matrices of perfect 2‐colorings for odd v. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 232–252, 2013     

A Reduction of the Anticoloring Problem to Connected Graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, for such colorings with many vertices colored in each color. We show that, to solve the anticoloring problem with two colors for general graphs, it suffices to solve it for connected graphs.     

Injective colorings of planar graphs with few colors

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥ 19 and maximum degree Δ are injectively Δ-colorable. We also show that all planar graphs of girth ≥ 10 are injectively (Δ+1)-colorable, that Δ+4 colors are sufficient for planar graphs of girth ≥ 5 if Δ is large enough, and that subcubic planar graphs of girth ≥ 7 are injectively 5-colorable.     

Anticoloring and separation of graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, such colorings with many vertices colored in each color. We deal with the anticoloring problem for planar graphs and, using Lipton and Tarjan’s separation algorithm, provide an algorithm with some bound on the error. We also show that, to solve the anticoloring problem for general graphs, it suffices to solve it for connected graphs.     

Gallai colorings of non-complete graphs

Gallai-colorings of complete graphs-edge colorings such that no triangle is colored with three distinct colors-occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result-any Gallai-colored complete graph has a monochromatic spanning tree-in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least (α(H)²+α(H)−1)⁻¹|V(H)| vertices, where α(H) is the independence number of H. In general, we show that if the edges of an r-uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H₁⊆H such that |V(H₁)|≥c|V(H)| where c depends only on F, r, and α(H).     

Circular list colorings of some graphs

The circular list coloring is a circular version of list colorings of graphs. Let χinc,l denote the circular choosability(or the circular list chromatic number). In this paper, the circular choosability of outer planar graphs and odd wheel is discussed.     

Non-intersecting perfect matchings in cubic graphs (Extended abstract)

A conjecture of G. Fan and A. Raspaud asserts that every bridgeless cubic graph contains three perfect matchings with empty intersection. We suggest a possible approach to problems of this type, based on the concept of a balanced join in an embedded graph. The method can be used to prove a special case of a conjecture of E. Máčajová and M. Škoviera on Fano colorings of cubic graphs.     

On Group Choosability of Total Graphs

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures. We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar graphs and near outerplanar graphs with maximum degree at least 4. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at most two.     

Hom complexes and hypergraph colorings

Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.     

On periodicity of perfect colorings of the infinite hexagonal and triangular grids

A coloring of vertices of a graph G is called r-perfect, if the color structure of each ball of radius r in G depends only on the color of the center of the ball. The parameters of a perfect coloring are given by the matrix A = (a _ij ) _i,j=1 ⁿ , where n is the number of colors and a _ij is the number of vertices of color j in a ball centered at a vertex of color i. We study the periodicity of perfect colorings of the graphs of the infinite hexagonal and triangular grids. We prove that for every 1-perfect coloring of the infinite triangular and every 1- and 2-perfect coloring of the infinite hexagonal grid there exists a periodic perfect coloring with the same matrix. The periodicity of perfect colorings of big radii have been studied earlier.     

A note on strong perfectness of graphs

In (strongly) perfect graphs, we define (strongly) canonical colorings; we show that for some classes of graphs, such colorings can be obtained by sequential coloring techniques. Chromatic properties ofP ₄-free graphs based on such coloring techniques are mentioned and extensions to graphs containing no inducedP ₅, orC ₅ are presented. In particular we characterize the class of graphs in which any maximal (or minimal) nodex in the vicinal preorder has the following property: there is either noP ₄ havingx as a midpoint or noP ₄ havingx as an endpoint. For such graphs, according to a result of Chvatal, there is a simple sequential coloring algorithm.     

Circular colorings of edge‐weighted graphs

The notion of (circular) colorings of edge‐weighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds to the traveling salesman problem (metric case). It also gives rise to a new definition of the chromatic number of directed graphs. Several basic results about the circular chromatic number of edge‐weighted graphs are derived. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 107–116, 2003     

Tverberg’s Theorem and Graph Coloring

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Δ, where Δ is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>Δ ² then the topological Tverberg theorem still works. It is conjectured that q>KΔ is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.     

Edge colorings of complete graphs without tricolored triangles

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.