A nonincreasing sequence π = ( d 1 , …,d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π . Given a graph H , a graphic sequence π is potentially H-graphic if π has a realization containing H as a subgraph. For graphs G 1 and G 2 , the potential-Ramsey number r pot ( G 1 , G 2 ) is the smallest integer k such that for every k -term graphic sequence π , either π is potentially G 1 -graphic or the complementary sequence $$\overline \pi = \left({k - 1 - {d_k}, \cdots k - 1 - {d_1}} \right)$$ π ¯ = ( k − 1 − d k , ⋯ k − 1 − d 1 ) is potentially G 2 -graphic. For $$0 \le k \le \left\lfloor {{t \over 2}} \right\rfloor $$ 0 ≤ k ≤ ⌊ t 2 ⌋ , denote $$K_t^{- k}$$ K t − k to be the graph obtained from K t by deleting k independent edges. If k = 0, Busch et al. (Graphs Combin., 30(2014)847–859) present a lower bound on r pot ( G, K t ) by using the 1-dependence number of G . In this paper, we utilize i -dependence number of G for i ≥ 1 to give a new lower bound on r pot ( G , $$K_t^{- k}$$ K t − k ) for any k with $$0 \le k \le \left\lfloor {{t \over 2}} \right\rfloor $$ 0 ≤ k ≤ ⌊ t 2 ⌋ . Moreover, we also determine the exact values of r pot ( K n , $$K_t^{- k}$$ K t − k ) for 1 ≤ k ≤ 2.

中文翻译：

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两个图的潜在拉姆齐数的新下界

非负整数的非递增序列 π = ( d 1 , …,dn ) 是图形序列，如果它可以通过 n 个顶点上的简单图 G 实现。在这种情况下，G 被称为 π 的实现。给定图 H ，如果 π 具有包含 H 作为子图的实现，则图形序列 π 可能是 H 图形的。对于图 G 1 和 G 2 ，势拉姆齐数 r pot ( G 1 , G 2 ) 是最小的整数 k 使得对于每个 k 项图形序列 π ， π 可能是 G 1 -graphic 或互补序列$$\overline \pi = \left({k - 1 - {d_k}, \cdots k - 1 - {d_1}} \right)$$ π¯ = ( k − 1 − dk , ⋯ k − 1 − d 1 ) 可能是 G 2 -图形。对于 $$0 \le k \le \left\lfloor {{t \over 2}} \right\rfloor $$ 0 ≤ k ≤ ⌊ t 2 ⌋ ，表示 $$K_t^{- k}$$ K t − k是通过删除 k 个独立边从 K t 获得的图。如果 k = 0，布施等人。(Graphs Combin., 30(2014)847–859) 通过使用 G 的 1 依赖数，给出了 r pot ( G, K t ) 的下限。在本文中，我们利用 G 的 i -dependence number for i ≥ 1 为任何具有 $$0 \ le k \le \left\lfloor {{t \over 2}} \right\rfloor $$ 0 ≤ k ≤ ⌊ t 2 ⌋ . 此外，我们还确定了 1 ≤ k ≤ 2 时 r pot ( K n , $$K_t^{- k}$$K t − k ) 的确切值。