Journal of Combinatorial Theory Series B最新文献

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

Let $X$ be a family of subsets of a finite set E. A matroid on E is called an $X$-matroid if each set in $X$ is a circuit. We develop techniques for determining when there exists a unique maximal $X$-matroid in the weak order poset of all $X$-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families $X$ and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.

Alon and Krivelevich proved that for every n-vertex subcubic graph H and every integer $q\ge 2$ there exists a (smallest) integer $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of H in which the length of every subdivision-path is divisible by q. Improving their superexponential bound, we show that $f(H,q)\le \frac{21}{2}qn+8n+14q$, which is optimal up to a constant multiplicative factor.

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(\mathrm{tw},\omega )$-bounded. The family of $(\mathrm{tw},\omega )$-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that $(\mathrm{tw},\omega )$-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent $(\mathrm{tw},\omega )$-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for $(\mathrm{tw},\omega )$-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.

Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition $T$ of a graph as the maximum independence number over all subgraphs of G induced by some bag of $T$. The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of $(\mathrm{tw},\omega )$-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor.

By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.

The Gyárfás-Sumner conjecture says that for every forest H and every integer k, if G is H-free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H, if G is H-free and does not contain $K_{t,t}$ as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H, the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer $d\ge 2$, if G is H-free and does not contain as a subgraph the complete d-partite graph with parts of cardinality t, then its chromatic number is at most polynomial in t.

We say a class $C$ of graphs is clean if for every positive integer t there exists a positive integer $w\left(t\right)$ such that every graph in $C$ with treewidth more than $w\left(t\right)$ contains an induced subgraph isomorphic to one of the following: the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, a subdivision of the $(t\times t)$-wall or the line graph of a subdivision of the $(t\times t)$-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H) is clean if and only if H is a forest whose components are subdivided stars.

Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η, a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G. This is of independent interest, and will be used in subsequent papers in this series.

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.

• 1.

  For $t\ge 2$, a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all $k,t$ and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some $t\ge 2$).

• 2.

  A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

We prove that for any $\epsilon >0$, for any large enough t, there is a graph that admits no $K_t$-minor but admits a $(\frac{3}{2}-\epsilon )t$-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.