A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a theorem which characterizes uniquely 2–list colorable graphs. We introduce the uniquely list chromatic number of a graph and make a conjecture about it which is a...

A graph is (k1, k2)-colorable if its vertex set can be partitioned into a graph with maximum degree at most k1 and and a graph with maximum degree at most k2. We show that every (C3, C4, C6)-free planar graph is (0, 6)-colorable. We also show that deciding whether a (C3, C4, C6)-free planar graph is (0, 3)-colorable is NP-complete.

This work studies the hardness of finding independent sets in hypergraphs which are either 2colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1−ε)-almost 2-colorable if removing an ε fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph...

A graph G is class II, if its chromatic index is at least ∆ + 1. Let H be a maximum ∆-edge-colorable subgraph of G. The paper proves best possible lower bounds for |E(H)| |E(G)| , and structural properties of maximum ∆-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with ∆ ≥ 3 can be extended to a maximum ∆-edge-colorable subgraph. Simple graph...

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study r-resiliently k-colorable graphs, which are those k-colorable graphs that remain k-colorable even after the addition of any r new edges. We prove lower...

A Steiner triple system of order v, STS(v), is an ordered pair S = (V,B), where V is a set of size v and B is a collection of triples of V such that every pair of V is contained in exactly one triple of B. A k-block coloring is a partitioning of the set B into k color classes such that every two blocks in one color class do not intersect. In this paper, we introduce a construction and use it to...

We give a new proof that the Petersen graph is not 3-edge-colorable. J. Petersen introduced the most well known graph, the Petersen graph, as an example of a cubic bridgeless graph that is not Tait colorable, i.e. it is not 3-edge-colorable. It is easy to see the equivalence between the following statements, but most proofs for each of them use a case by case argument [1]. Theorem 1 For the Pet...

A graph is (2, 1)-colorable if it allows a partition of its vertices into two classes such that both induce graphs with maximum degree at most one. A non-(2, 1)-colorable graph is minimal if all proper subgraphs are (2, 1)colorable. We prove that such graphs are 2-edge-connected and that every edge sits in an odd cycle. Furthermore, we show properties of edge cuts and particular graphs which ar...

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G − e is not a uniquely k-colorable graph for any edge e ∈ E(G). Mel’nikov and Steinberg [L. S. Mel’nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] as...

A graph G is (k, 0)-colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 k+2 is (k, 0)-colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)-colorable. On the othe...