Polychromatic colorings of arbitrary rectangular partitions

A general (rectangular) partition is a partition of a rectangle into an arbitrary number of non-overlapping subrectangles. This paper examines vertex 4-colorings of general partitions where every subrectangle is required to have all four colors appear on its boundary. It is shown that there exist general partitions that do not admit such a coloring. This answers a question of Dimitrov et al. [D. Dimitrov, E. Horev, R. Krakovski, Polychromatic colorings of rectangular partitions, Discrete Mathematics 309 (2009) 2957-2960]. It is also shown that the problem to determine if a given general partition has such a 4-coloring is NP-Complete. Some generalizations and related questions are also treated.     

A note on panchromatic colorings

This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r\le c\frac{n}{lnn}$ then all bounds have a type $A_1(n,lnn,r){\left(\frac{r}{r?1}\right)}^n\le p(n,r)\le A_2(n,r,lnr){\left(\frac{r}{r?1}\right)}^n$, where $A_1$, $A_2$ are some algebraic fractions. The main result is a new lower bound on $p(n,r)$ when $r$ is at least $c\sqrt{n}$; we improve an upper bound on $p(n,r)$ if $n=o\left(r^{3∕2}\right)$.Also we show that $p(n,r)$ has upper and lower bounds depending only on $n∕r$ when the ratio $n∕r$ is small, which cannot be reached by the previous probabilistic machinery.Finally we construct an explicit example of a hypergraph without panchromatic coloring and with ${(\frac{r}{r?1}+o\left(1\right))}^n$ edges for $r=o\left(\sqrt{\frac{n}{lnn}}\right)$.     

Oriented list colorings of graphs

A 2‐assignment on a graph G = (V,E) is a collection of pairs L(v) of allowed colors specified for all vertices v ∈V. The graph G (with at least one edge) is said to have oriented choice number 2 if it admits an orientation which satisfies the following property: For every 2‐assignment there exists a choice c(v)∈L(v) for all v ∈V such that (i) if c(v) = c(w), then vw ∉ E, and (ii) for every ordered pair (a,b) of colors, if some edge oriented from color a to color b occurs, then no edge is oriented from color b to color a. In this paper we characterize the following subclasses of graphs of oriented choice number 2: matchings; connected graphs; graphs containing at least one cycle. In particular, the first result (which implies that the matching with 11 edges has oriented choice number 2) proves a conjecture of Sali and Simonyi. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 217–229, 2001     

Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

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).     

Locally restricted colorings

We consider the following problem: given suitable integers χ and p, what is the smallest value ρ such that, for any graph G with chromatic number χ and any vertex coloring of G with at most χ+p colors, there is a vertex v such that at least χ different colors occur within distance ρ of v? Let ρ(χ,p) be this value; we show in particular that ρ(χ,p)?⌈p/2⌉+1 for all χ,p. We give the exact value of ρ when p=0 or χ?3, and (χ,p)=(4,1) or (4,2).     

Generating infinitely many satisfactory colorings of positive integers

An $n$-satisfactory coloring of the $n$-smooth integers is an assignment of $n$ colors to the positive integers whose prime factors are at most $n$ so that for each such $m$, the integers $m,2m,\dots ,nm$ receive different colors. In this note, we give a short proof that infinitely many 6-satisfactory colorings of the 6-smooth integers exist and show how the technique of the proof can be applied more generally, including for $n\in \{8,10,12\}$.     

Size of monochromatic components in local edge colorings

An edge coloring of a graph is a local r coloring if the edges incident to any vertex are colored with at most r distinct colors. We determine the size of the largest monochromatic component that must occur in any local r coloring of a complete graph or a complete bipartite graph.     

Bounds on vertex colorings with restrictions on the union of color classes

A proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets. In this paper, given a positive integer j and a family ? of connected graphs, we consider proper colorings in which we require that the union of any j color classes induces a subgraph which has no copy of any member of ?. This generalizes some well‐known types of proper colorings like acyclic colorings (where j = 2 and ?is the collection of all even cycles) and star colorings (where j = 2 and ?consists of just a path on 4 vertices), etc. For this type of coloring, we obtain an upper bound of O(d^{(k ? 1)/(k ? j)}) on the minimum number of colors sufficient to obtain such a coloring. Here, d refers to the maximum degree of the graph and k is the size of the smallest member of ?. For the case of j = 2, we also obtain lower bounds on the minimum number of colors needed in the worst case. As a corollary, we obtain bounds on the minimum number of colors sufficient to obtain proper colorings in which the union of any j color classes is a graph of bounded treewidth. In particular, using O(d^8/7) colors, one can obtain a proper coloring of the vertices of a graph so that the union of any two color classes has treewidth at most 2. We also show that this bound is tight within a multiplicative factor of O((logd)^1/7). We also consider generalizations where we require simultaneously for several pairs (j_i, ?_i) (i = 1, …, l) that the union of any j_i color classes has no copy of any member of ?_i and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011     

Vertex colorings of graphs without short odd cycles

Motivated by the work of Ne?et?il and Rödl on “Partitions of vertices” we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r‐coloring of the vertices of H yields a monochromatic and induced copy of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 255‐264, 2011     

Acyclic systems of representatives and acyclic colorings of digraphs

A natural digraph analog of the graph theoretic concept of “an independent set” is that of “an acyclic set of vertices,” namely a set not spanning a directed cycle. By this token, an analog of the notion of coloring of a graph is that of decomposition of a digraph into acyclic sets. We extend some known results on independent sets and colorings in graphs to acyclic sets and acyclic colorings of digraphs. In particular, we prove bounds on the topological connectivity of the complex of acyclic sets, and using them we prove sufficient conditions for the existence of acyclic systems of representatives of a system of sets of vertices. These bounds generalize a result of Tardos and Szabó. We prove a fractional version of a strong‐acyclic‐coloring conjecture for digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 177–189, 2008     

Polychromatic colorings of bounded degree plane graphs

A polychromatic k‐coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, one seeks the maximum number k such that G admits a polychromatic k ‐coloring. In this paper, it is proven that every connected plane graph of order at least three, and maximum degree three, other than K₄ or a subdivision of K₄ on five vertices, admits a 3‐coloring in the regular sense (i.e., no monochromatic edges) that is also a polychromatic 3‐coloring. Our proof is constructive and implies a polynomial‐time algorithm. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 269‐283, 2009     

A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t,n be integers, tnΔ. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t>Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t=n=Δ and G is regular.     

Edge‐colorings avoiding rainbow stars

We consider an extremal problem motivated by a article of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge‐colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given , we look for n‐vertex graphs that admit the maximum number of r‐edge‐colorings such that at most colors appear in edges incident with each vertex, that is, r‐edge‐colorings avoiding rainbow‐colored stars with t edges. For large n, we show that, with the exception of the case , the complete graph is always the unique extremal graph. We also consider generalizations of this problem.     

Some new bounds for the maximum number of vertex colorings of a (v,e)-graph

Let f(v, e, λ) denote the maximum number of proper vertex colorings of a graph with v vertices and e edges in λ colors. In this paper we present some new upper bounds for f(v, e, λ). In particular, a new notion of pseudo-proper colorings of a graph is given, which allows us to significantly improve the upper bounds for f(v, e, 3) given by Lazebnik and Liu in the case where e > v²/4. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 115–128, 1998