Graph coloring with decision diagrams

Mathematical Programming - We introduce an iterative framework for solving graph coloring problems using decision diagrams. The decision diagram compactly represents all possible color classes,...     

Random graphs with monochromatic triangles in every edge coloring

We prove that for every r ? 2 there is C > 0 such that if p Cn^?1/2 then almost surely every r-coloring of the edges of the binomial random graph K(n, p) results in a monochromatic triangle. © 1994 John Wiley & Sons, Inc.     

Graph products and monochromatic multiplicities

Arcane two-edge-colourings of complete graphs were described in [13], in which there are significantly fewer monochromaticK _r 's than in a random colouring (so disproving a conjecture of Erds [2]). Jagger, ovíek and Thomason [7] showed that the same colourings have fewer monochromaticG's than do random colourings for any graphG containingK ₄.The purpose of this note is to point out that these colourings are not as obscure as might appear. There is in fact a large, natural and easily described class of colourings of the above kind; the specific examples used in [13] and [7] fall into this class.     

Graph coloring and monotone functions on posets

The purpose of this note is to point out a relationship between graph coloring and monotone functions defined on posets. This relationship permits us to deduce certain properties of the chromatic polynomial of a graph.     

Linear coloring of planar graphs with large girth

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of the graph G is the smallest number of colors in a linear coloring of G. In this paper we prove that every planar graph G with girth g and maximum degree Δ has if G satisfies one of the following four conditions: (1) g≥13 and Δ≥3; (2) g≥11 and Δ≥5; (3) g≥9 and Δ≥7; (4) g≥7 and Δ≥13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.     

Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a(G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if g(G)≥cΔ r log(Δ2/r) then χa(G)≤Δ + r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a(G) = Δ if Δ≥4 and g(G)≥4; or Δ≥3 and g(G)≥5.     

Covering complete partite hypergraphs by monochromatic components

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson (1971)) states that $k$-partite intersecting hypergraphs have transversals of at most $k?1$ vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in Gyárfás (1977): if the edges of a complete graph $K$ are colored with $k$ colors then the vertex set of $K$ can be covered by at most $k?1$ sets, each forming a connected graph in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: it was proved in Király (2013) that in every $k$-coloring of the edges of the $r$-uniform complete hypergraph $K^r$ ($r\ge 3$), the vertex set of $K^r$ can be covered by at most $?k∕r?$ sets, each forming a connected hypergraph in some color.Here we investigate the analogue problem for complete $r$-uniform $r$-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of every color used in the coloring. We propose the following analogue of Ryser’s conjecture.In every spanning $(r+t)$-coloring of the edges of a complete $r$-uniform $r$-partite hypergraph, the vertex set can be covered by at most $t+1$ sets, each forming a connected hypergraph in some color.We show that the conjecture (if true) is best possible. Our main result is that the conjecture is true for $1\le t\le r?1$. We also prove a slightly weaker result for $t\ge r$, namely that $t+2$ sets, each forming a connected hypergraph in some color, are enough to cover the vertex set.To build a bridge between complete $r$-uniform and complete $r$-uniform $r$-partite hypergraphs, we introduce a new notion. A hypergraph is complete $r$-uniform $(r,?)$-partite if it has all $r$-sets that intersect each partite class in at most $?$ vertices (where $1\le ?\le r$).Extending our results achieved for $?=1$, we prove that for any $r\ge 3,2\le ?\le r,k\ge 1+r??$, in every spanning $k$-coloring of the edges of a complete $r$-uniform $(r,?)$-partite hypergraph, the vertex set can be covered by at most $1+?\frac{k?r+??1}{?}?$ sets, each forming a connected hypergraph in some color.     

Graph coloring in a class of parallel local algorithms

A way of improving the performance of a distributed algorithm is rendering a coloring strategy into an algorithm known as efficient in the nondistributed case. In this paper it is shown that certain sequential coloring algorithm heuristics like largest-first (LF), smallest-last (SL), and saturation largest-first (SLF), as applied to some classes of graphs and to special cases of vertex coloring in distributed algorithms, produce an optimal or near-optimal coloring.     

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.     

There is no fast method for finding monochromatic complete subgraphs

A general theorem on the complexity of a class of recognition problems is proved. As a particular case the following result is given: There is no algorithm which, for any 2-coloration of the infinite complete graph, can produce a monochromatic subgraph of k vertices within $\text{2}{}^{\mathrm{<mtext>k</mtext><mtext>2</mtext>}}$ steps (at each step the color of an arbitrary edge is questioned).     

Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given ?>0,

logL(r,f)>(1−?)logM(r,f)