Upper embeddability and connectivity of graphs

In this paper, we prove that every cyclically 4-edge-connected graph is upper-embeddable. The proof is based on the study of the effect on the maximum genus of a graph when one of its subgraphs is collapsed to a vertex.     

A CHARACTERIZATION OF EMBEDDABILITY OF GRAPHS ON SURFACES

In this article the Cn-graphs are introduced, by which a characterization of the embeddability of a graph on either an orientable surface or a non-orientable surface is provided.     

A sufficient condition on upper embeddability of graphs

In this paper we mainly prove that let G be a(k + 1)-edge-connected simple graph of order n with girth g.Then G is upper embeddable if for any independent set I(G) = {vi | 1 i k2 + 2},k = 0,1,2 and the lower bound is tight.     

On embeddability and stresses of graphs

Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K ₅-minor freeness guarantees the stress freeness. More generally, we prove that every K _r+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed. Supported by an I.S.F. grant.     

Surface embeddability of graphs via homology

This paper presents a characterization of the embeddability on a surface of genus arbitrarily given for a graph. Its specific case for the surface of genus zero leads to the famous planarity theorems given independently by Whitney via duality, MacLane via cycle basis and Lefschetz via double covering at a time.     

关于图的上可嵌入性的一个注记

任韩和李刚在图的最大亏格综述一文"Survey of maximum genus of graphs" [J East China NormUniv Natur Sci, Sep. 2010, No. 5, 1-13] 中,全面地阐述了近30 年来关于图的最大亏格及其相关问题所取得的进展,并提出了如下两个猜想:
猜想1 设G 为简单连通图, 且G 的每条边含在一个三角形K₃ 中, 则G 是上可嵌入的.
猜想2 设c 为任意的正数, 则存在一个自然数N(c), 使得对每一个图G, 若G 的点数n ≥ N(c), 且最小度δ(G) ≥ cn, 则G 是上可嵌入的.
本文的主要工作是否定上述两个猜想, 同时探讨上述猜想成立的条件且得了一些新结果, 并提出有关进一步研究的问题.     

Signed graphs with at most three eigenvalues

Czechoslovak Mathematical Journal - We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or...     

Bidirectional retracting-free double tracings and upper embeddability of graphs

We characterize those finite connected graphs which admit a closed walk such that each edge is traversed once in each direction and such that no edge is succeeded by the same edge in the opposite direction. This solves an old problem of O. Ore.     

A full characterization of invariant embeddability of unimodular planar graphs

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.     

Q-integral graphs with edge-degrees at most five

We consider the problem of determining the Q-integral graphs, i.e. the graphs with integral signless Laplacian spectrum. We find all such graphs with maximum edge-degree 4, and obtain only partial results for the next natural case, with maximum edge-degree 5.     

Generating cycles in graphs with at most one end

Answering a question of Halin, we prove that in a 3‐connected graph with at most one end the cycle space is generated by induced non‐separating cycles. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 342–349, 2003     

All hypoenergetic graphs with maximum degree at most 3

The energy E(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. A connected graph G of order n is said to be hypoenergetic if E(G)<n. All connected hypoenergetic graphs with maximum degree Δ3 have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph K_2,3 is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by Majstorović et al. has been confirmed.     

Full orientability of graphs with at most one dependent arc

Suppose that D is an acyclic orientation of a graph G. An arc of D is dependent if its reversal creates a directed cycle. Let () denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G. We call Gfully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying . We show that a connected graph G is fully orientable if . This generalizes the main result in Fisher et al. [D.C. Fisher, K. Fraughnaugh, L. Langley, D.B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71 (1997) 73-78].     

Maximal independent sets in graphs with at most r cycles

We find the maximum number of maximal independent sets in two families of graphs. The first family consists of all graphs with n vertices and at most r cycles. The second family is all graphs of the first family which are connected and satisfy n ≥ 3r. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 270–282, 2006     

Extremal graphs of diameter at most 6 after deleting any vertex

Suppose G is a graph of n vertices and diameter at most d having the property that, after deleting any vertex, the resulting subgraph has diameter at most 6. Then G contains at least max{n, (4n - 8)/3} edges if 4 ≤ d ≤ 6.