arXiv subjects
Publications and source records attributed to Zdenek Dvorak.
We show that Erdős-Rényi random graphs $G(n,p)$ with constant density $p<1$ have correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger's conjecture for correspondence coloring. The proof follows from a simple sufficient condition for correspondence colorability in terms of the numbers of independent sets.
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size D can be colored using at most D+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for D=16 and D=17.
We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that the sum of lengths of (>=5)-faces of G is at most linear in g+t+c-1.
We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way.
We give a natural sufficient condition for an intersection graph of compact convex sets in R^d to have a balanced separator of sublinear size. This condition generalizes several previous results on sublinear separators in intersection graphs. Furthermore, the argument used to prove the existence of sublinear separators is based on a connection with generalized coloring numbers which has not been previously explored in geometric settings.
We prove that if every subgraph of a graph $G$ has a balanced separation of order at most $a$ then $G$ has treewidth at most $15a$. This establishes a linear dependence between the treewidth and the separation number.
Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.
A graph H is t-apex if H-X is planar for some subset X of V(H) of size t. For any integer t>=0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t+3)-connected H-minor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t>=1, there exists a t-apex graph H such that testing (t+4)-colorability of (t+2)-connected H-minor-free graphs is NP-complete. Similarly, the size of the lists cannot be decreased (unless P=NP), since for every t>=1, testing (t+3)-list-colorability of (t+3)-connected K_{t+4}-minor-free graphs is NP-complete.
We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.
By Brook's Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an indepdendent set of size at least n/Delta+k has a kernel of size O(k).
We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.
We give a new proof of the fact that every planar graph is 5-choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings.
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.
We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.
Let Q_n be the graph of n times n times n cube with all non-decreasing diagonals (including the facial ones) in its constituent unit cubes. Suppose that a subset S of V(Q_n) separates the left side of the cube from the right side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We use a generalization of this claim to prove that the vertex set of Q_n cannot be partitioned to two parts, each of them inducing a subgraph of bounded tree-width.
A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author.