Combinatorica最新文献

In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in ({mathbb {R}}^3) that can be associated to the cycle diagram of a permutation. We show that Woo’s characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham’s inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.

Reidl et al. (Eur J Comb 75:152–168, 2019) characterized graph classes of bounded expansion as follows: A class ({mathcal {C}}) closed under subgraphs has bounded expansion if and only if there exists a function (f:{mathbb {N}} rightarrow {mathbb {N}}) such that for every graph (G in {mathcal {C}}), every nonempty subset A of vertices in G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is at most f(r) |A|. When ({mathcal {C}}) has bounded expansion, the function f(r) coming from existing proofs is typically exponential. In the special case of planar graphs, it was conjectured by Sokołowski (Electron J Comb 30(2):P2.3, 2023) that f(r) could be taken to be a polynomial. In this paper, we prove this conjecture: For every nonempty subset A of vertices in a planar graph G and every nonnegative integer r, the number of distinct intersections between A and a ball of radius r in G is ({{,mathrm{{mathcal {O}}},}}(r^4 |A|)). We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs.

Given a graph G with a set F(v) of forbidden values at each (v in V(G)), an F-avoiding orientation of G is an orientation in which (deg ^+(v) not in F(v)) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if (|F(v)| < frac{1}{2} deg (v)) for each (v in V(G)), then G has an F-avoiding orientation, and they showed that this statement is true when (frac{1}{2}) is replaced by (frac{1}{4}). In this paper, we take a step toward this conjecture by proving that if (|F(v)| < lfloor frac{1}{3} deg (v) rfloor ) for each vertex v, then G has an F-avoiding orientation. Furthermore, we show that if the maximum degree of G is subexponential in terms of the minimum degree, then this coefficient of (frac{1}{3}) can be increased to (sqrt{2} - 1 - o(1) approx 0.414). Our main tool is a new sufficient condition for the existence of an F-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.

Let (D=(V,A)) be a digraph. For an integer (kge 1), a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer (tau ge 1), suppose (d_A^+(U)+(frac{tau }{k}-1)d_A^-(U)ge tau ) for all (Usubsetneq V, Une emptyset ), where (d_A^+(U)) and (d_A^-(U)) denote the number of arcs in A leaving and entering U, respectively. Let ({mathcal {C}}) be a crossing family over ground set V, and let (f:{mathcal {C}}rightarrow {mathbb {Z}}) be a crossing submodular function such that (f(U)ge frac{k}{tau }(d_A^+(U)-d_A^-(U))) for all (Uin {mathcal {C}}). Then D has a k-arc-connected flip J such that (f(U)ge d_J^+(U)-d_J^-(U)) for all (Uin {mathcal {C}}). The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is (tau )-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

The induced size-Ramsey number (hat{r}_text {ind}^k(H)) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., (hat{r}_text {ind}^k(C_n)le Cn) for some (C=C(k)). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that (hat{r}_text {ind}^k(C_n)le O(k^{102})n) when n is even, thus obtaining only a polynomial dependence of C on k. We also prove (hat{r}_text {ind}^k(C_n)le e^{O(klog k)}n) for odd n, which almost matches the lower bound of (e^{Omega (k)}n). Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies (hat{r}^k(C_n)=e^{O(k)}n) for odd n. This substantially improves the best previous result of (e^{O(k^2)}n), and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

A family (mathcal {F} subset mathcal {P}(n)) is r-wise k-intersecting if (|A_1 cap dots cap A_r| ge k) for any (A_1, dots , A_r in mathcal {F}). It is easily seen that if (mathcal {F}) is r-wise k-intersecting for (r ge 2), (k ge 1) then (|mathcal {F}| le 2^{n-1}). The problem of determining the maximum size of a family (mathcal {F}) that is both (r_1)-wise (k_1)-intersecting and (r_2)-wise (k_2)-intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255–1266, 2019). They proved the surprising result that, for ((r_1,k_1) = (3,1)) and ((r_2,k_2) = (2,32)) then this maximum is at most (2^{n-2}), and conjectured the same holds if (k_2) is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for ((r_1,k_1) = (3,1)) and ((r_2,k_2) = (2,3)) for all n.

We answer a question posed by Gallai in 1969 concerning criticality in Sperner’s lemma, listed as Problem 9.14 in the collection of Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). Sperner’s lemma states that if a labelling of the vertices of a triangulation of the d-simplex (Delta ^d) with labels (1, 2, ldots , d+1) has the property that (i) each vertex of (Delta ^d) receives a distinct label, and (ii) any vertex lying in a face of (Delta ^d) has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For (dle 2), it is not difficult to show that for every facet (sigma ), there exists a labelling with the above properties where (sigma ) is the unique rainbow facet. For every (dge 3), however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai’s question as a corollary. The construction is based on the properties of a 4-polytope which had been used earlier to disprove a claim of Motzkin on neighbourly polytopes.

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph (K_{k,k}) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is (Oleft( n^{2 - 1/k}right) ). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that (k ge d ge 2). A remarkable result of Fox et al. (J. Eur. Math. Soc. (JEMS) 19:1785–1810, 2017) with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of (K_{k,k}) as a subgraph must be (Oleft( n^{2 - 1/d}right) ). This theorem is sharp when (k=d=2), because by design any (K_{2,2})-free graph automatically has VC-dimension at most 2, and there are well-known examples of such graphs with (Omega left( n^{3/2}right) ) edges. However, it turns out this phenomenon no longer carries through for any larger d. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of (K_{k,k}) and VC-dimension at most d is (o(n^{2-1/d})), for every (k ge d ge 3).

Given an integer (k>4) and a graph H, we prove that, assuming P(ne ) NP, the List-k -Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if either every component of H is a path on at most three vertices, or removing the isolated vertices of H leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all (kge 1).