In this paper we prove that every planar graph without 4, 5 and 8-cycles is 3-colorable.

Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, k-colorability of a graph can be characterized in terms of whether its graph polynomial is contained in a certain ideal. In this paper, we interpret unique colorability in an analogous manner and prove an algebraic characterization for uniquely k-colo...

It is shown that the difference between the chromatic number χ and the fractional chromatic number χf can be arbitrarily large in the class of uniquely colorable, vertex transitive graphs. For the lexicographic product G ◦ H it is shown that χ(G ◦ H) ≥ χf (G)χ(H). This bound has several consequences, in particular it unifies and extends several known lower bounds. Lower bounds of Stahl (for gen...

A proper vertex coloring of a graph G = (V , E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v) | v ∈ V } of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V . If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V , then G is acyclically k-choosable...

A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let Hk(n,m) be a random k-uniform hypergraph on n vertices formed by picking m edges uniformly, independently and with replacement. It is easy to show that if r ≥ rc = 2k−1 ln 2− (ln 2)/2, then with high probability Hk(n,m = rn) is not 2-colorable. We complement this observatio...

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1 ∈ IG,k for a certain ideal IG,k ⊆ k[x1, . . . , xn]. In this paper, we extend this result by proving a general decomposition theorem for IG,k . This theorem allows us to giv...

Motivated by a conjecture of Grünbaum and problem Katona, Kostochka, Pach, Stechkin, both dealing with non-Hamiltonian $n$-vertex graphs their $(n-2)$-cycles, we investigate $K_2$-Hamiltonian graphs, i.e., in which the removal any pair adjacent vertices yields Hamiltonian graph. In this first part, prove structural properties show that there exist infinitely many cubic 3-edge-colorable non-3-ed...

A signed graph, S, is colorable if its point set can be partitioned into subsets such that all positive lines join points of the same subset and all negative lines join points of different subsets. S is uniquely colorable if there is only one such partition. Developed in this note is a new matrix, called the type matrix of S, which provides a classification of the way pairs of points are joined...

The talk deals with combinatorial problems concerning colorings of non-uniform hyper-graphs. Let H = (V, E) be a hypergraph with minimum edge-cardinality n. We show that if H is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and e∈E r 1−|e| c √ n, for some absolute constant c > 0, then H is r-colorable. We also obtain a stronger result for triangle-free simpl...