Cycles through all finite vertex sets in infinite graphs

For two integers m,n with m ≤ n, an [m, n]-factor F in a graph G is a spanning subgraph of G with m ≤ dF(v) ≤ n for all v ∈ V(F). In 1996, H. Enomoto et al. proved that every 3-connected planar graph G with dG(v) ≥ 4 for all v ∈ V(G) contains a [2,3]-factor. In this paper we extend their result to all 3-connected locally finite infinite planar graphs containing no unbounded faces.

Optimal identifying codes of two families of Cayley graphs

Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre's semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.

By the method of nonstandard analysis, the definition of *-finite graph is given, and necessary and sufficient conditions of *-finite graph are obtained. Further, by the Transfer Principle, we apply the theory of finite graph to *-finite graph, embed given infinite graph into some *-finite graph, and finally obtain the related results of infinite graph.

All finite planar graphs are subgraphs of certain planar Cayley graphs of free groups; outerplanar graphs are subgraphs of certain outerplanar Cayley graphs of free groups. It is shown for these particular planar presentations of free groups that special planar is equivalent with outerplanar.KeywordsCayley GraphCyclic PermutationSpecial PlanarOuterplanar GraphLocal FinitenessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let G(V, E) be a finite or infinite (locally finite) graph with the $$CD\psi (m,-K)$$ condition for some constants $$m>0,K\ge 0$$, and some $$C^1$$, concave function $$\psi : (0,+\infty )\rightarrow {\mathbf {R}}$$. In this paper, we firstly prove that if $$u:V\rightarrow (0,+\infty )$$ is a solution to the equation $$\triangle _{\mu } u=-\lambda u$$, then $$\lambda \le \frac{mK}{4\psi '(1)}$$, and u satisfies an elliptic gradient estimate. We can recover a result derived in Wang and Zhou (Archiv der Mathematik 109:383–391, 2017) if we choose $$\psi =\sqrt{\cdot }$$. Secondly, we establish a general gradient estimate for the linear heat equation $$(\triangle _{\mu }-\partial _t) u=0$$; this estimate includes the Davies, Hamilton, Bakry–Qian and Li–Xu’s estimates; these four type estimates had been established on the manifold. We can derive gradient estimates on a finite graph with the same types as in the manifold case if we choose $$\psi =\ln {(\cdot )}$$. As by-products, we get corresponding Harnack inequalities.

The μ-Calculus Alternation Depth Hierarchy is infinite over finite planar graphs

A finite graph G is a pair (V, E), where V is a finite set of vertices and E is a subset of V (2), the set of unordered pairs of elements of V. The elements of E are called edges. (This is what graph theorists call a simple graph.

Counting 1-factors in infinite graphs

On the Metric Dimension of Infinite Graphs

Let G be a graph and let s be a vertex of G. We consider the structure of the set of all lifts of two edges incident with s that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least $$(\deg (s)-1)/2$$(deg(s)-1)/2 disjoint pairs that lift. We consider the lifting graph: its vertices are the edges incident with s, two being adjacent if they form a liftable pair. We have three main results, the first two with the same hypotheses as for Mader's Theorem. (i) Let F be a subset of the edges incident with s. We show that F is independent in the lifting graph of G if and only if there is a single edge-cut C in G of size at most $$r+1$$r+1 containing all the edges in F, where r is the maximum number of edge-disjoint paths from a vertex (not s) in one component of $$G-C$$G-C to a vertex (not s) in another component of $$G-C$$G-C. (ii) In the k-lifting graph, two edges incident with s are adjacent if their lifting leaves the resulting graph with the property that any two vertices different from s are joined by k pairwise edge-disjoint paths. If both $$\deg (s)$$deg(s) and k are even, then the k-lifting graph is a connected complete multipartite graph. In all other cases, there are at most two components. If there are exactly two components, then each component is a complete multipartite graph. If $$\deg (s)$$deg(s) is odd and there are two components, then one component is a single vertex. (iii) Huck proved that if k is odd and G is $$(k+1)$$(k+1)-edge-connected, then G is weakly k-linked (that is, for any k pairs $$\{x_i,y_i\}$${xi,yi}, there are k edge-disjoint paths $$P_i$$Pi, with $$P_i$$Pi joining $$x_i$$xi and $$y_i$$yi). We use our results to extend a slight weakening of Huck's theorem to some infinite graphs: if k is odd, every $$(k+2)$$(k+2)-edge-connected, locally finite, 1-ended, infinite graph is weakly k-linked.

It is well known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalize the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.

The study of a variation of the marking game, in which the first player marks vertices and the second player marks edges of an undirected graph was proposed by Bartnicki et al. (Electron J Combin 15:R72, 2008). In this game, the goal of the second player is to mark as many edges around an unmarked vertex as possible, while the first player wants just the opposite. In this paper, we prove various bounds for the corresponding graph invariant, the vertex-edge coloring number \({\text {col}}_\mathrm{ve}(G)\) of a graph G. In particular, every (finite or infinite) graph G whose edges can be oriented in such a way that the maximum out-degree is bounded by an integer d has \({\text {col}}_\mathrm{ve}(G)\le d+2\). We investigate this invariant in (classes of) planar graphs, including some infinite lattices. We present a close connection between the vertex-edge coloring number of a graph G and the game coloring number of the subdivision graph S(G). In our main result, we bound the vertex-edge coloring number in complete graphs from below and from above, and while \({\text {col}}_\mathrm{ve}(K_n)\le \lceil \log _2{n}\rceil +2\), the difference between the upper and the lower bound is roughly \(\log _2(\log _2 n)\). The latter results are, in fact, true for any multigraph whose underlying graph is \(K_n\).

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then 'V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and ' i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollob\'as's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.