Related papers: Counterexamples to Thomassen's conjecture on decom…

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is…

A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number,…

The problem of 2-coloring uniform hypergraphs has been extensively studied over the last few decades. An n-uniform hypergraph is not 2-colorable if its vertices can't be colored with two colors, Red and Blue, such that every hyperedge…

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no…

Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations.

For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a…

We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its…

In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof…

We introduce a new invariant of a cubic graph - its regular colouring defect - which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a…

We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree at least 13.

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…

The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances).…

In this short note, we present a purely entropic proof that in a $3$-edge-colored simple graph with $R$ red edges, $G$ green edges, and $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$.

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly…

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. We pose a new conjecture which implies Thomassen's conjecture.…

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order…

A packing $(1^{\ell}, 2^k)$-coloring of a graph $G$ is a partition of $V(G)$ into $\ell$ independent sets and $k$ $2$-packings (whose pairwise vertex distance is at least $3$). The square coloring of planar graphs was first studied by…

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs…