A Sane Proof that COLk COL3

Journal of Combinatorial Theory, Series B, 1990

Arxiv preprint cs/0609083, 2006

Discrete Mathematics, 2009

We show that if H is an odd-cycle, or any non-bipartite graph of girth 5 and maximum degree at most 3, then planar H-COL is NP-complete.

Information Processing Letters, 2002

A partition of the vertices of a graph G into k pairwise disjoint sets V 1 , . . . , V k is called an (r 1 , . . . , r k )-subcoloring if the subgraph G i of G induced by V i , 1 i k, consists of disjoint complete subgraphs, each of which has cardinality no more than r i . Due to Erdős and Albertson et al., independently, every cubic (i.e., 3-regular) graph has a (2, 2)-subcoloring. Albertson et al. then asked for cubic graphs having (1, 2)-subcolorings. We point out in this paper that this question is algorithmically difficult by showing that recognizing (1, 2)-subcolorable cubic graphs is NP-complete, even when restricted to triangle-free planar graphs. 

2018

A $(a,b)$-coloring of a graph $G$ associates to each vertex a set of $b$ colors from a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoints. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, we prove necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices. Other more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice. Computations on millions of such graphs generated randomly show that our tools allow to find (in linear time) a $(9,4)$-coloring for each of them. Although there remain few graphs for which our tools are not sufficient for finding a $(9,4)$-coloring, we believe that pursuing our method can lead to a solution of the conjecture of McDiarmid-Reed.

Journal of Discrete Algorithms, 2012

SIAM Journal on Discrete Mathematics

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry and are well studied in graph theory. Here we study the natural problem of the conflict-free chromatic number \chi CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then \chi CF (G) \leq k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k \in \{ 1, 2, 3\} , it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.

2001

Discrete Mathematics, 1990

For integers k 2 1 and m 2 2 a (k, m)-colouring of a graph G is a colouring of the vertices of G in k colours such that no m-clique of G is monocoloured.

2010

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.

AKCE International Journal of Graphs and Combinatorics, 2020

A total coloring of a graph G is an assignment of colors to the vertices and the edges such that (i) no two adjacent vertices receive same color, (ii) no two adjacent edges receive same color, and (iii) if an edge e is incident on a vertex v, then v and e receive different colors. The least number of colors sufficient for a total coloring of graph G is called its total chromatic number and denoted by v 00 ðGÞ: An adjacent vertex distinguishing (AVD)-total coloring of G is a total coloring with the additional property that for any adjacent vertices u and v, the set of colors used on the edges incident on u including the color of u is different from the set of colors used on the edges incident on v including the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of G, v 00 a ðGÞ is the minimum number of colors required for a valid AVD-total coloring of G. It is conjectured that v 00 ðGÞ DðGÞ þ 2, which is known as total coloring conjecture and is one of the famous open problems. A graph for which the total coloring conjecture holds is called totally colorable graph. The problem of deciding whether v 00 ðGÞ ¼ DðGÞ þ 1 or v 00 ðG ¼ DðGÞ þ 2 for a totally colorable graph G is called the classification problem for total coloring. However, this classification problem is known to be NP-hard even for bipartite graphs. In this paper, we give a sufficient condition for a bipartite biconvex graph G to have v 00 ðGÞ ¼ DðGÞ þ 1: Also, we propose a linear time algorithm to compute the total chromatic number of chain graphs, a proper subclass of biconvex graphs. We prove that the total coloring conjecture holds for the central graph of any graph. Finally, we obtain the AVD-total chromatic number of central graphs for basic graphs such as paths, cycles, stars and complete graphs.

Lecture Notes in Computer Science, 2006

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to O(n 4/11) colors. This is the first combinatorial improvement of Blum's O(n 3/8) bound from FOCS'90. Like Blum's algorithm, our new algorithm composes nicely with recent semi-definite approaches. The current best bound is O(n 0.2072) colors by Chlamtac from FOCS'07. We now bring it down to O(n 0.2038) colors.

Journal of Graph Theory, 2019

A colouring of a graph G = (V, E) is a function c : V → {1, 2,. . .} such that c(u) = c(v) for every uv ∈ E. A k-regular list assignment of G is a function L with domain V such that for every u ∈ V , L(u) is a subset of {1, 2,. .. } of size k. A colouring c of G respects a k-regular list assignment L of G if c(u) ∈ L(u) for every u ∈ V. A graph G is k-choosable if for every k-regular list assignment L of G, there exists a colouring of G that respects L. We may also ask if for a given k-regular list assignment L of a given graph G, there exists a colouring of G that respects L. This yields the k-Regular List Colouring problem. For k ∈ {3, 4} we determine a family of classes G of planar graphs, such that either k-Regular List Colouring is NP-complete for instances (G, L) with G ∈ G, or every G ∈ G is k-choosable. By using known examples of non-3-choosable and non-4-choosable graphs, this enables us to classify the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and no 5-cycles. We also classify the complexity of k-Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.

Information Processing Letters, 2008

We proved that every planar graph in which the cycles of length 3, 4, and 5 are at distance at least 4 from each other is 3choosable.