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Injective edge-coloring of claw-free subcubic graphs
An injective edge-coloring of a graph G is an edge-coloring of G such that any two edges that are at distance 2 or in a common triangle receive...
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On the Alon–Tarsi number of semi-strong product of graphs
The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The Alon–Tarsi number AT ( G ) of a graph G is...
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Absence of zeros implies strong spatial mixing
In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs that is...
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On (3, r)-Choosability of Some Planar Graphs
There are many refinements of list coloring, one of which is the choosability with union separation. Let k , s be positive integers and let G be a...
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Incidence Coloring of Outer-1-planar Graphs
A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge....
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Multiple DP-Coloring of Planar Graphs Without 3-Cycles and Normally Adjacent 4-Cycles
The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvořák and Postle (J. Combin. Theory Ser. B 129 , 38–54,...
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Coloring of Hypergraphs
Hypergraphs are discrete structures that generalize graphs in a very natural way. While in a graph every edge is incident with exactly two vertices,...
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Colorings and Orientations of Graphs
Colorings and orientations of graphs are related in different ways, but the deepness of these relations is notwell understood. In this chapter we...
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Degree Bounds for the Chromatic Number
Until the 1940s, coloring theory focused almost exclusively on coloring of maps. However, already in 1879 A. B. Kempe suggested coloring of abstract...
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Critical Graphs with few Edges
This chapter is concerned with the minimum number ext(k,n) of edges in k-critical graphswith n vertices. Brooks’ theorem says that 2ext(k,n) ≥ (k...
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On Decidability of Hyperbolicity
We prove that a wide range of coloring problems for graphs on surfaces can be resolved by inspecting a finite number of configurations.
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Degeneracy and Colorings
As discussed in the previous chapter, every graph 𝐺 satisfies 𝜒(𝐺) ≤ col(𝐺) ≤ Δ(𝐺) + 1. However, for many graph classes, the difference between...
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Attempting perfect hypergraphs
We study several extensions of the notion of perfect graphs to k -uniform hypergraphs. One main definition extends to hypergraphs the notion of...
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Coloring Graphs on Surfaces
The coloring of maps or, equivalently, the coloring of graphs on surfaces is one of the most popular topics in graph coloring theory. This chapter...