For a simple graph G=(V,E), a proper total weighting is a mapping w:V∪E→R such that for every edge uv∈E, w(u)+∑e∋uw(e)≠w(v)+∑e∋vw(e). The graph G is (2,2)-choosable if, for any list assignment L that assigns to each z in V∪E a set L(z) of two real numbers, there exists a proper total weighting w with w(z)∈L(z) for every z∈V∪E. Wong and Zhu, and independently Przybyło and Woźniak conjectured that every simple graph is (2,2)-choosable. This conjecture remains open.For a set {a,b}⊂R, its span is |b−a|. We call G uniform-span (2,2)-choosable if for every list assignment L where all lists L(z) have size two and have a common span, there exists a proper total weighting w with w(z)∈L(z) for all z∈V∪E. In this paper, we present a novel lemma and perform comprehensive enhancements to our previous algorithm. These contributions enable us to prove that every graph is uniform-span (2,2)-choosable, and thus admits a proper total weighting using any prescribed pair {a,b}⊂R (including {1,2}). This confirms and extends the 1-2 conjecture.
An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. We prove for r ∈{4,5}, every projective planar r-graph with no Petersen-minor is r-edge colorable.
We study sufficient conditions for the generic rigidity of a graph G expressed in terms of (i) its minimum degree δ(G), or (ii) the parameter η(G)=minuv∉E(deg(u)+deg(v)). For each case, we seek the smallest integers f(n,d) (resp. g(n,d)) such that every n-vertex graph G with δ(G)≥f(n,d) (resp. η(G)≥g(n,d)) is rigid in Rd.Krivelevich, Lew, and Michaeli conjectured that there is a constant K>0 such that f(n,d)≤n2+Kd for all pairs n,d. We give an affirmative answer to this conjecture by proving that K=1 suffices. For n≥29d, we obtain the exact result f(n,d)=⌈n+d−22⌉.Next, we prove that g(n,d)≤n+3d for all pairs n,d, and establish g(n,d)=n+d−2 when n≥d(d+2). For d=2,3, we determine the exact values of f(n,d) and g(n,d) for all n, confirming another conjecture of Krivelevich, Lew, and Michaeli in these low-dimensional special cases.As an application, we prove that the Erdős-Rényi random graph G(n,1/2) is a.a.s. rigid in Rd for d=d(n)∼732n. This result provides the first linear lower bound for d(n), and it answers a question of Peled and Peleg.
We prove two new results on linkages and orientations in infinite graphs. In both theorems, we provide bounds within 1 from the bounds suggested in the respective conjectures.In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s1,t1}, …, {sk,tk} in G, not necessarily distinct, there are pairwise edge-disjoint paths P1,…,Pk in G, with Pi joining si and ti. In 1991, Huck proved that the conclusion holds if G is finite and (k+1)-edge-connected. We prove that Huck's theorem holds also for all infinite graphs, extending and improving a result of Ok, Richter and Thomassen for 1-ended, locally finite graphs.In 1960 Nash-Williams suggested that his theorem, that 2k-edge-connected finite graphs admit a k-arc-connected orientation, also holds for infinite graphs. We prove that an edge-connectivity of 2k+1 is sufficient for k-arc-connected orientations of infinite graphs.A novel key tool in the proofs is the linking fan proposition proved in Section 3. We first use it to prove the linkage theorem, then we apply it in the last section to prove a result, similar to a result of Thomassen, on the existence of 2k-edge-connected finite immersions in (2k+1)-edge-connected infinite graphs. This latter immersion theorem is needed for the proof of the orientation theorem.
The problem of chromatic thresholds seeks minimum degree conditions that ensure that dense H-free graphs have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem further requires that the homomorphic image itself is H-free. The purpose of this paper is two-fold.First, we define a generalized notion of threshold which encapsulates and interpolates chromatic and homomorphism thresholds via the theory of VC-dimension. Our first main result shows a smooth transition between these two thresholds when varying the restrictions on homomorphic images. In particular, we proved that for t≥s≥3 and ε>0, if G is an n-vertex Ks-free graph with VC-dimension d and δ(G)≥((s−3)(t−s+2)+1(s−2)(t−s+2)+1+ε)n, then G is homomorphic to a Kt-free graph H with |H|=Os,t,d,ε(1). Moreover, we construct graphs showing that this minimum degree condition is optimal. This extends and unifies the results of Thomassen, Łuczak and Thomassé, and Goddard, Lyle and Nikiforov, and provides a deeper insight into the existence of homomorphic images with various properties.Second, we introduce the blowup threshold δB(H) as the infimum α such that every n-vertex maximal H-free graph G with δ(G)≥αn is a blowup of some F with |F|=Oα,H(1). This notion strengthens homomorphism thresholds. While the homomorphism thresholds for odd cycles remain unknown, we prove that δB(C2k−1)=12k−1 for any integer k≥2. This strengthens the result of Ebsen and Schacht and answers a question of Schacht and shows that, in sharp contrast to the chromatic thresholds, 0 is an accumulation point for blowup thresholds.Our proofs mix tools from VC-dimension theory and an iterative refining process to find the desired homomorphic images, and draw a connection to a problem concerning codes on graphs.
