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1. A Unified Proof of Conjectures on Cycle Lengths in Graphs

   https://doi.org/10.1093/imrn/rnaa324  

   Gao, Jun; Huo, Qingyi; Liu, Chun-Hung; Ma, Jie (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $$G$$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $$k$$; in addition, if $$G$$ is $$2$$-connected and non-bipartite, then it contains cycles of all lengths modulo $$k$$. 2. For all $$k\geq 3$$, every $$k$$-connected graph contains a cycle of length zero modulo $$k$$. 3. Every $$3$$-connected non-bipartite graph with minimum degree at least $k+1$ contains $$k$$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $$k$$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible. 

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2. Simple Topological Drawings of k-Planar Graphs.

   https://doi.org/10.1007/978-3-030-68766-3_31  

   Hoffmann, Michael; Liu, Chih-Hung; Reddy, Meghana M.; Toth, Csaba D. (February 2021, Graph Drawing and Network Visualization)

   null (Ed.)

   Every finite graph admits a simple (topological) drawing, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, k-planar graphs are those graphs that can be drawn so that every edge has at most k crossings (i.e., they admit a k-plane drawing). It is known that for k≤3 , every k-planar graph admits a k-plane simple drawing. But for k≥4 , there exist k-planar graphs that do not admit a k-plane simple drawing. Answering a question by Schaefer, we show that there exists a function Open image in new window such that every k-planar graph admits an f(k)-plane simple drawing, for all Open image in new window. Note that the function f depends on k only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every 4-planar graph admits an 8-plane simple drawing. 

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3. On 2-fold graceful graphs

   Bunge, R; Cornett, M; El-Zanati, S; Jeffries, J; Rattin, E. (January 2020, Journal of Combinatorial Mathematics and Combinatorial Computing)

   null (Ed.)

   A long-standing conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree $$T$$ with $$n$$~edges, it is conjectured that there exists a labeling $$f\colon V(T) \to \{0,1,\ldots,n\}$$ such that the set of induced edge labels $$\bigl\{ |f(u)-f(v)| : \{u,v\}\in E(T) \bigr\}$$ is exactly $$\{1,2,\ldots,n\}$$. We extend this concept to allow for multigraphs with edge multiplicity at most~$$2$$. A \emph{2-fold graceful labeling} of a graph (or multigraph) $$G$$ with $$n$$~edges is a one-to-one function $$f\colon V(G) \to \{0,1,\ldots,n\}$$ such that the multiset of induced edge labels is comprised of two copies of each element in $$\bigl\{ 1,2,\ldots, \lfloor n/2 \rfloor \bigr\}$$ and, if $$n$$ is odd, one copy of $$\bigl\{ \lceil n/2 \rceil \bigr\}$$. When $$n$$ is even, this concept is similar to that of 2-equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length $$n \not\equiv 1 \pmod{4}$$, and complete bipartite graphs admit 2-fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is $$2$$-fold graceful. 

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4. Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance

   https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.58  

   Ashvinkumar, Vikrant; Assadi, Sepehr; Deng, Chengyuan; Gao, Jie; Wang, Chen (January 2023, APPROX-RANDOM)

   Megow, Nicole; Smith, Adam (Ed.)

   Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced. We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation. To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest. 

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5. The Implicit Graph Conjecture is False

   https://doi.org/10.1109/FOCS54457.2022.00109  

   Hatami, Hamed; Hatami, Pooya (October 2022, Annual Symposium on Foundations of Computer Science)

   An efficient implicit representation of an n-vertex graph G in a family F of graphs assigns to each vertex of G a binary code of length O(log n) so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most 2^O(n log(n)) graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length n^Ω(1). 

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