In 2009, Belford and Sieben introduced the concept of graph rubbling, which added the rubbling move to the well-known graph pebbling game. While a pebbling move removes two pebbles from one vertex and places a new pebble on a neighboring vertex, a rubbling move takes one pebble from each of two distinct neighbors of some vertex v and places one new pebble on v. The optimal rubbling number of a graph is the minimum number of pebbles needed to reach each vertex using some combination of pebbling and rubbling moves. We introduce the strict optimal rubbling number ρstr(G), which we define to be the minimum number of pebbles needed to reach each vertex using only rubbling moves. We then determine the value of ρstr for and handful of families of graphs, highlighting its similarities and differences to the optimal rubbling number.

Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447 ). In this new setting, each vertex v in some subset of V(G) has a request for a certain color r(v) in its list of colors L(v). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant ε >0 such that any graph G in some graph class 𝒞 satisfies at least ε proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G ∈𝒞 on vertex set V, with any list assignment L of size k for each vertex, and for every R ⊆ V and a request vector (r(v): v∈ R, r(v) ∈ L(v)) , there exists an L-coloring of G satisfying at least ε |R| requests. If this is true, then 𝒞 is called ε -flexible for lists of size k. Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021 ) introduced the notion of weak flexibility, where R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists ε (b)>0 so that the class of planar graphs without K_4, C_5 , C_6 , C_7, B_b is weakly ε (b) -flexible for lists of size 4 (here K_n , C_n and B_n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K_4, C_5 , C_6 , C_7, B_5 is ε -flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

The Pentagon Problem of Erdős problem asks to find an n-vertex triangle-free graph that is maximizing the number of 5-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladký, Král’, Norin, and Razborov. Recently, Palmer suggested a more general problem of maximizing the number of 5-cycles in K k + 1-free graphs. Using flag algebras, we show that every K k + 1-free graph of order n contains at most 1 10 k 4 ( k 4 − 5 k 3 + 10 k 2 − 10 k + 4 ) n 5 + o ( n 5 ) copies of C 5 for any k ≥ 3, with the Turán graph being the extremal graph for large enough n.

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex v ∈ V, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.