Dense $H$-Free Graphs are Almost $(\chi(H)-1)$-Partite

Tutte proved that every planar 4‐connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex‐deleted subgraphs are hamiltonian. We here prove that if in a planar ‐vertex graph with minimum degree at least 4 at least vertex‐deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every there exists a nonhamiltonian polyhedral ‐vertex graph with minimum degree at least 4 containing hamiltonian vertex‐deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex‐deleted subgraphs as well as Bondy's meta‐conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex‐deleted subgraphs are traceable, must itself be traceable.

Partitions of graphs and multigraphs under degree constraints

With the rapid expansion of social networks, efficiently mining and analyzing massive graph data has become a fundamental challenge in social network research. Graph partitioning plays a pivotal role in enhancing the performance of such analyses. However, conventional graph partitioning methods predominantly rely on local structural information and often overlook the rich attribute information associated with vertices in social network graphs. To overcome this limitation, this paper introduces GP-DQN (Graph Partitioning via Double Deep Q-Network), a large-scale graph partitioning algorithm that jointly considers structural correlations, attribute disparities among user vertices, and partition load balancing. GP-DQN encodes partition load metrics and vertex attributes into vector representations and employs a Graph Convolutional Network (GCN) to aggregate both vertex features and neighborhood structures, thereby improving the accuracy and scalability of the partitioning process. A tailored reward function is designed to guide partitioning actions, where a Double Deep Q-Network (DDQN) predicts the expected partitioning rewards based on GCN-extracted features for assigning each vertex to different partitions. The partitioning strategy is iteratively optimized using both immediate and expected rewards, ultimately achieving balanced load distribution while minimizing the number of edge cuts. Experimental results demonstrate that GP-DQN produces well-balanced partitions with significantly fewer edge cuts, leading to enhanced computational efficiency within each partition.

The famous Dirac's theorem states that for each every ‐vertex graph with minimum degree has a Hamiltonian cycle. When , this cannot be guaranteed, but the existence of some other specific subgraphs can be provided. Gargano, Hammar, Hell, Stacho, and Vaccaro proved that every connected ‐vertex graph with contains a spanning spider , that is, a spanning tree with at most one vertex of degree at least 3. Later, Chen, Ferrara, Hu, Jacobson, and Liu proved the stronger (and exact) result that for every connected ‐vertex graph with contains a spanning broom , that is, a spanning spider obtained by joining the center of a star to an endpoint of a path. They also showed that a 2‐connected graph with and some additional properties contains a spanning jellyfish , which is a graph obtained by gluing the center of a star to a vertex in a cycle disjoint from that star. Note that every spanning jellyfish contains a spanning broom. The goal of this paper is to prove an exact Ore‐type bound which guarantees the existence of a spanning jellyfish: We prove that if is a 2‐connected graph on vertices such that every nonadjacent pair of vertices satisfies , then has a spanning jellyfish. As corollaries, we obtain strengthenings of two results by Chen et al.: a minimum degree condition guaranteeing the existence of a spanning jellyfish, and an Ore‐type sufficient condition for the existence of a spanning broom. The corollaries are sharp for infinitely many . One of the main ingredients of our proof is a modification of the Hopping Lemma due to Woodall.

Graph partitions with minimum degree constraints

For a ‐vertex graph and an ‐vertex graph , an ‐tiling in is a collection of vertex‐disjoint copies of in . For , the ‐independence number of , denoted , is the largest size of a ‐free set of vertices in . In this article, we discuss Ramsey–Turán‐type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal ‐tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.

C. Thomassen and M. Szegedy proved the existence of a functionf(s, t) such that the points of anyf(s, t)-connected graph have a decomposition into two non-empty sets such that the subgraphs induced by them ares-connected andt-connected, respectively. We prove, thatf(s, t) ≦ 4s+4t − 13 and examine a similar problem for the minimum degree.

The clique removal lemma says that for every and , there exists some so that every ‐vertex graph with fewer than copies of can be made ‐free by removing at most edges. The dependence of on in this result is notoriously difficult to determine: it is known that must be at least super‐polynomial in , and that it is at most of tower type in . We prove that if one imposes an appropriate minimum degree condition on , then one can actually take to be a linear function of in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

\n Vertex-centric graph processing systems such as Pregel, PowerGraph,\n or GraphX recently gained popularity due to their superior\n performance of data analytics on graph-structured data. These\n systems exploit the graph structure to improve data access locality\n during computation, making use of specialized graph partitioning\n algorithms. Recent partitioning techniques assume a uniform and\n constant amount of data exchanged between graph vertices (i.e.,\n uniform vertex traffic) and homogeneous underlying network costs.\n However, in real-world scenarios vertex traffic and network costs\n are heterogeneous. This leads to suboptimal partitioning decisions\n and inefficient graph processing. To this end, we designed GrapH,\n the first graph processing system using vertex-cut graph\n partitioning that considers both, diverse vertex traffic and\n heterogeneous network, to minimize overall communication costs. The\n main idea is to avoid frequent communication over expensive network\n links using an adaptive edge migration strategy. Our evaluations\n show an improvement of 60% in communication costs compared to\n state-of-the-art partitioning approaches.\n

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.

We obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of , using a palette of size , a.a.s. admits a rainbow copy of any given bounded‐degree tree on at most vertices, where is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded‐degree almost‐spanning prescribed trees in , where is independent of . Given an ‐vertex graph with minimum degree at least , where is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph , using colors, where is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded‐degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that , where is independent of , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with as above, we prove that a uniform coloring of using colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

The balanced graph partitioning consists in dividing the vertices of an undirected graph into a given number of subsets of approximately equal size, such that the number of edges crossing the subsets is minimized. In this work, we present a multilevel memetic algorithm for this NP-hard problem that relies on a powerful grouping recombination operator and a dedicated local search procedure. The proposed operator tends to preserve the backbone with respect to a set of parent individuals, i.e. the grouping of vertices which is same throughout each parent individual. Although our approach requires significantly longer computing time compared to some current state-of-art graph partitioning algorithms such as SCOTCH, METIS, CHACO, JOSTLE, etc., it competes very favorably with these approaches in terms of solution quality. Moreover, it easily reaches or improves on the best partitions ever reported in the literature.

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.