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Let G be an edge-colored copy of Kn, where each color appears on at most n/2 edges (the edgecoloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth [4] conjectured that every properly edge-colored Kn (n ≥ 6 and even) using exactly n − 1 colors has n/2 edge-disjoint rainbow spanning trees, and they proved there are at least two edge-disjoint rainbow spanning trees. Kaneko, Kano, and Suzuki [13] strengthened the conjecture to include any proper edge-coloring of Kn, and they proved there are at least three edgedisjoint rainbow spanning trees. Akbari and Alipouri [1] showed that each Kn that is edge-colored such that no color appears more than n/2 times contains at least two rainbow spanning trees.We prove that if n ≥ 1, 000, 000 then an edge-colored Kn, where each color appears on at most n/2 edges, contains at least n/(1000 log n) edge-disjoint rainbow spanning trees.

Given a family scriptF and a host graph H, a graph G⊆H is scriptF‐saturated relative to H if no subgraph of G lies in scriptF but adding any edge from E(H)−E(G) to G creates such a subgraph. In the scriptF‐saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in scriptF, until G becomes scriptF‐saturated relative to H. They aim to maximize or minimize the length of the game, respectively; sat gfalse(scriptF;Hfalse) denotes the length under optimal play (when Max starts). Let scriptO denote the family of odd cycles and Tn the family of n‐vertex trees, and write F for scriptF when F={F}. Our results include sat gfalse(scriptO;Knfalse)=⌊n2⌋false⌈n2false⌉, sat gfalse(Tn;Knfalse)=false(0ptn−22false)+1 for n≥6, sat gfalse(K1,3;Knfalse)=2⌊n2⌋ for n≥8, and sat gfalse(P4;Knfalse)∈{⌊4n5⌋,⌈4n5⌉} for n≥5. We also determine sat gfalse(P4;Km,nfalse); with m≥n, it is n when n is even, m when n is odd and m is even, and m+⌊n/2⌋ when mn is odd. Finally, we prove the lower bound sat gfalse(C4;Kn,nfalse)≥121n13/12−Ofalse(n35/36false). The results are very similar when Min plays first, except for the P4‐saturation game on Km,n.

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph G is the maximum density of an independent set in G. Lih, Liu, and Zhu [31] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs.We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.

In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph G and a function f : V (G) → N, the graph is f -paintable if there is an algorithm to produce a proper coloring when each vertex v is allowed to be presented at most f (v) times.The sum-paintability. . , H k are the blocks of G. Also, adding an ear of length ℓ to G adds 2ℓ − 1 to the sum-paintability, when ℓ ≥ 3.Strengthening a result of Berliner et al., we prove χ sp (K 2,r ) = 2r+min{l+m : lm > r}. The generalized theta-graph Θ ℓ 1 ,...,ℓ k consists of two vertices joined by internally disjoint paths of lengths ℓ 1 , . . . , ℓ k . A book is a graph of the form Θ 1,2,...,2 , denoted B r when there are r internally disjoint paths of length 2. We prove χ sp (B r ) = 2r + min l,m∈N {l + m : m(l − m) + m 2 > r}. We use these results to determine the sumpaintability for all generalized theta-graphs.