Related papers: Wheel-free planar graphs
This is the first in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs…
Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to…
A path in a vertex-colored graph is called \emph{conflict free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be \emph{conflict-free vertex-connected} if any two vertices of the graph are…
A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is…
We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This…
Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar…
In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial…
Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a…
We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a…
A \emph{$k$-track} layout of a graph consists of a vertex $k$ colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. A \emph{$k$-queue} layout of a graph consists of a…
Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon)…
A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles…
A clique-coloring of a given graph $G$ is a coloring of the vertices of $G$ such that no maximal clique of size at least two is monocolored. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a…
In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable.
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…
An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex colouring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph colouring was independently…
We study a conjecture due to Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht stating that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor…
We prove that, for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$ in which every $F$-free induced subgraph has chromatic…
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large…
We prove that a graph $G$ contains no induced $5$-vertex path and no induced complement of a $5$-vertex path if and only if $G$ is obtained from $5$-cycles and split graphs by repeatedly applying the following operations: substitution,…