Journal of Combinatorial Theory Series B

For any small positive real ε and integer t > 1 ε, we build a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2 ( 1 − ε ) t. In particular, this shows that the twin-width is sometimes exponential in the treewidth,...

We describe an infinite family of strongly 2-connected oriented graphs (that is, directed graphs with no multiple arcs) containing no arc whose reversal decreases the number of directed cycles.

We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.

We investigate the question how ‘small’ a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a ...

The proper orientation number χ → ( G ) of a graph G is the minimum k such that there exists an orientation of the edges of G with all vertex-outdegrees at most k and such that for any adjacent vertices, the outdegrees are different. Two major ...

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has chromatic number at most ( C + o ( 1 ) ) Δ / log ⁡ Δ for some universal constant C > 0. Using the entropy compression method, Molloy proved that one can in fact take C ...

Let G = ( V , E ) be a simple graph with n vertices and m edges, P ( G , k ) be the chromatic polynomial of G, and P ( G , L ) be the number of L-colorings of G for any k-assignment L. In this article, we show that when k ≥ m − 1 ≥ 3, P ( G , L ) ...

In this paper we investigate the bipartite analogue of the strong Erdős-Hajnal property. We prove that for every forest H and every τ with 0 < τ ≤ 1, there exists ε > 0, such that if G has a bipartition ( A , B ) and does not contain H as an ...

We present a systematic investigation into how tree-decompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be ...

Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G ∖ S belongs to G. We denote by A k ( G ) the set of all graphs that are k-apices of G. We prove that every graph ...

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs ( G , γ ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γ-...

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 ⋅ 4 ω − 1. The ...

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ > 0 and integers ℓ , r and n with n ∈ r N, is it true that there exists an α > 0 such that every n-vertex graph G with δ ( G ) ≥ max ⁡ { 1 2 , r − ...

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set ...

We prove that for every t ∈ N there is a constant γ t such that every graph with twin-width at most t and clique number ω has chromatic number bounded by 2 γ t log 4 t + 3 ⁡ ω. In other words, we prove that graph classes of bounded twin-width are ...

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least C k 2 has a subdivision of K k, the complete graph on k vertices. We ...

For integer n > 0, let f ( n ) be the number of rows of the largest all-0 or all-1 square submatrix of M, minimized over all n × n 0/1-matrices M. Thus f ( n ) = O ( log ⁡ n ). But let us fix a matrix H, and define f H ( n ) to be the same, ...

Grosu (2016) [11] asked if there exist an integer r ≥ 3 and a finite family of r-graphs whose Turán density, as a real number, has (algebraic) degree greater than r − 1. In this note we show that, for all integers r ≥ 3 and d, there exists a ...