Related papers: Triangle-free Uniquely 3-Edge Colorable Cubic Grap…

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured…

We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not…

For a non-decreasing positive integer sequence $S = (s_{1}, \dots, s_{k})$, an $S$-packing edge-coloring of a graph $G$ is a partition of the edge set of $G$ into subsets $E_{1}, \dots, E_{k}$ such that for each $1 \leq i \leq k$, the…

Bonamy et al. (2023) proved that an optimal edge coloring of a simple triangle--free graph $G$ can be reached from any given proper edge coloring of $G$ through a series of Kempe changes. We show that a small modification of their proof…

In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a three-edge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the…

Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every…

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…

In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $\chi$-bounded but we allow here the existence of…

We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vuskovic.

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin asked whether…

In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.

Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with $n$ vertices and at most $2^{15n/\log_2 n}$ 3-colorings. We improve…

Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$…

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is…

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs,…

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…

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least…

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct…