ABSTRACT We prove that every ‐vertex planar graph with no triangle sharing an edge with a 4‐cycle has independence ratio for . This result implies that the same bound holds for 4‐cycle‐free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5‐cycle. For the latter case we strengthen the bound to .

ABSTRACT For two given graphs and , a graph is said to be weakly ‐saturated if is a spanning subgraph of which has no copy of as a subgraph and one can add all edges in to in some order so that a new copy of is created at each step. The weak saturation number is the minimum number of edges of a weakly ‐saturated graph. In this paper, we deal with the relation between and , where denotes the Erdős–Rényi random graph and denotes the complete graph on vertices. For any graph and constant , we prove that with high probability. Also, for some graphs , including complete graphs, complete bipartite graphs, and connected graphs with minimum degree or , it is shown that there exists an such that, for every with high probability.

ABSTRACT 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.

ABSTRACTBrooks' Theorem is a fundamental result on graph colouring, stating that the chromatic number of a graph is almost always upper bounded by its maximal degree. Lovász showed that such a colouring may then be computed in linear time when it exists. Many analogues are known for variants of (di)graph colouring, notably for list‐colouring and partitions into subgraphs with prescribed degeneracy. One of the most general results of this kind is due to Borodin, Kostochka, and Toft, when asking for classes of colours to satisfy ‘variable degeneracy’ constraints. An extension of this result to digraphs has recently been proposed by Bang‐Jensen, Schweser, and Stiebitz, by considering colourings as partitions into ‘variable weakly degenerate’ subdigraphs. Unlike earlier variants, there exists no linear‐time algorithm to produce colourings for these generalisations. We introduce the notion of (variable) bidegeneracy for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of ‐dicolouring, where is a vector of functions, and an ‐dicolouring requires vertices coloured to induce a ‘strictly‐‐bidegenerate’ subdigraph. We prove an analogue of Brooks' theorem for ‐dicolouring, generalising the result of Bang‐Jensen et al., and earlier analogues in turn. Our new approach provides a linear‐time algorithm that, given a digraph , either produces an ‐dicolouring of , or correctly certifies that none exist. This yields the first linear‐time algorithms to compute (di)colourings corresponding to the aforementioned generalisations of Brooks' theorem. In turn, it gives an unified framework to compute such colourings for various intermediate generalisations of Brooks' theorem such as list‐(di)colouring and partitioning into (variable) degenerate sub(di)graphs.

ABSTRACTA graph drawn on the plane is called 1‐plane if each edge is crossed at most once by another edge. In this paper, we show that every 4‐edge‐connected 1‐plane graph has a connected spanning plane subgraph. We also show that there exist infinitely many 4‐connected 1‐plane graphs that have no 2‐connected spanning plane subgraphs. Moreover, we consider the condition of and such that every ‐connected 1‐plane graph has an ‐connected spanning plane subgraph.