A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γ R (G) of G is the minimum of v∈V (G) f (v) over such functions. Let G be a connected n-vertex graph. We prove that γ R (G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γ R (G)+γ R (G) and γ R (G)γ R (G), improving known results for domination number. We prove that γ R (G) ≤ 8n/11 when δ(G) ≥ 2 and n ≥ 9, and this is sharp. IntroductionAccording to [6], Constantine the Great (Emperor of Rome) issued a decree in the 4th century A.D. for the defense of his cities. He decreed that any city without a legion stationed to secure it must neighbor another city having two stationed legions. If the first were attacked, then the second could deploy a legion to protect it without becoming vulnerable itself.The objective, of course, is to minimize the total number of legions needed. The problem generalizes to arbitrary graphs. A Roman dominating function (RDF) on a graph G is a vertex labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. For an RDF f , let V i (f ) = {v ∈ V (G) : f (v) = i}. In the context of a fixed RDF, we suppress the argument and simply write V 0 , V 1 , and V 2 . Since this partition determines f , we can equivalently write f = (V 0 , V 1 , V 2 ). The weight w(f ) of an RDF f is v∈V (G) f (v), which equals |V 1 | + 2|V 2 |. The Roman domination number γ R (G) is the minimum weight of an RDF of G. Thus, γ R (G) is the minimum number of legions needed to protect cities whose adjacency graph is G.

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S, T) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.

Let G be a weighted graph in which each vertex initially has weight 1. A total acquisition move transfers all the weight from a vertex u to a neighboring vertex v, under the condition that before the move the weight on v is at least as large as the weight on u. The (total) acquisition number of G, written a t (G), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves.Among connected n-vertex graphs, a t (G) is maximized by trees. The maximum is Θ( √ n lg n) for trees with diameter 4 or 5. It is ⌊(n + 1)/3⌋ for trees with diameter between 6 and 2 3 (n + 1), and it is ⌈(2n − 1 − D)/4⌉ for trees with diameter D when 2 3 (n + 1) ≤ D ≤ n − 1. We characterize trees with acquisition number 1, which permits testing a t (G) ≤ k in time O(n k+2 ) on trees.If G = C 5 , then min{a t (G), a t (G)} = 1. If G has diameter 2, then a t (G) ≤ 32 ln n ln ln n; we conjecture a constant upper bound. Indeed, a t (G) = 1 when G has diameter 2 and no 4-cycle, except for four graphs with acquisition number 2.Deleting one edge of an n-vertex graph cannot increase a t by more than 6.84 √ n, but we construct an n-vertex tree with an edge whose deletion increases it by more than 1 2 √ n. We also obtain multiplicative upper bounds under products.

We consider the following question of Bollobá s: given an rcoloring of E(K n ), how large a k-connected subgraph can we find using at most s colors? We provide a partial solution to this problem when s 5 1 (and n is not too small), showing that when r 5 2 the answer is nÀ2k12, when r 5 3 the answer is b(nÀk)/2c11 or d(nÀk)/2e11, and when rÀ1 is a prime power then the answer lies between n/(rÀ1)À11(k 2 Àk)r and (nÀk1 1)/(rÀ1)1r. The case sZ2 is considered in a subsequent paper (Liu et

We consider the following question of Bollobás: given an r-colouring of E(K n ), how large a k-connected subgraph can we find using at most s colours? We provide a partial solution to this problem when s = 1 (and n is not too small), showing that when r = 2 the answer is n − 2k + 2, when r = 3 the answer is ⌊ n−k 2 ⌋ + 1 or ⌈ n−k 2 ⌉ + 1, and when r − 1 is a prime power then the answer lies between n r−1 − 11(k 2 − k)r and n−k+1 r−1 + r. The case s 2 is considered in a subsequent paper [6], where we also discuss some of the more glaring open problems relating to this question.

Let D(H ) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H ) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs K s,t for a fixed s and large t. Myers proved upper bounds on D(K s,t ) and made a conjecture on the order of magnitude of D(K s,t ) for a fixed s and t → ∞. He also found exact values for D(K 2,t ) for an infinite series of t. In this paper, we confirm the conjecture of Myers and find asymptotically (in s) exact bounds on D(K s,t ) for a fixed s and large t.

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