Journal of Combinatorial Theory Series B最新文献

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number $\chi \left(G\right)$ is upper bounded by a function of its clique number $\omega \left(G\right)$.

Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have $\chi \left(G\right)\le {\left(\omega \right(G\left)\right)}^3+\omega \left(G\right)$.

Robertson and Seymour constructed for every graph G a tree-decomposition that efficiently distinguishes all the tangles in G. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical.

The key ingredient is an axiomatisation of ‘local properties’ of tangles. Generalisations to locally finite graphs and matroids are also discussed.

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and $S\subseteq V\left(G\right)$ takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S.

We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows:

Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.

Let M be an excluded minor for the class of $P$-representable matroids for some partial field $P$, let N be a 3-connected strong $P$-stabilizer that is non-binary, and suppose M has a pair of elements $\{a,b\}$ such that $M﹨a,b$ is 3-connected with an N-minor. Suppose also that $\left|E\right(M\left)\right|\ge \left|E\right(N\left)\right|+11$ and $M﹨a,b$ is not N-fragile. In the prequel to this paper, we proved that $M﹨a,b$ is at most five elements away from an N-fragile minor. An element e in a matroid $M^{\prime }$ is N-essential if neither $M^{\prime }/e$ nor $M^{\prime }﹨e$ has an N-minor. In this paper, we prove that, under mild assumptions, $M﹨a,b$ is one element away from a minor having at least $r\left(M\right)-2$ elements that are N-essential.

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. We prove that, given $k,r>0$, there exists a k-connected common graph with chromatic number at least r. The result is built upon the recent breakthrough of Kráľ, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.

We find the exact value of the Ramsey number $R(C_{2\ell },K_{1,n})$, when ℓ and $n=O\left({\ell }^{10/9}\right)$ are large. Our result is closely related to the behaviour of Turán number $\mathrm{ex}(N,C_{2\ell })$ for an even cycle whose length grows quickly with N.

This paper proves the following result: If G is a planar graph and L is a 4-list assignment of G such that $\left|L\right(x)\cap L(y\left)\right|\le 2$ for every edge xy, then G is L-colourable. This answers a question asked by Kratochvíl et al. (1998) [10].

We study relation between two interpretations of the Tutte polynomial of a matroid perspective $M_1\to M_2$ on a set E given with a linear ordering <. A well known interpretation uses internal and external activities on a family $B(M_1,M_2)$ of the sets independent in $M_1$ and spanning in $M_2$. Recently we introduced another interpretation based on a family $D(M_1,M_2;<)$ of “cyclic bases” of $M_1\to M_2$ with respect to <. We introduce a one-to-one correspondence between $B(M_1,M_2)$ and $D(M_1,M_2;<)$ that also generates a relation between the interpretations of the Tutte polynomial of a matroid perspective and corresponds with duality.

A graph H is ubiquitous if every graph G that for every natural number n contains n vertex-disjoint H-minors contains infinitely many vertex-disjoint H-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe ($\delta \left(G\right)\ge {10}^{{10}^{10}}$), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of the same length, s say, along with s vertex-disjoint paths of the same length³ which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this paper, we give a proof of this conjecture by building a pillar (algorithmically) in sublinear expanders.