Spanning tree congestion of rook's graphs

Abstract

Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G joining the two components of T − e. The congestion of T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion over all its spanning trees. In this paper, we determine the spanning tree congestion of the rook's graph K m K n for any m and n.

On spanning tree congestion of graphs

“…It is evident that K d n is d(n − 1)-regular. The exact value of stc(K 2 n ) is known [6]. Also, stc(K d 2 ) is determined asymptotically [8].…”

“…The spanning tree congestion of graphs has been studied intensively [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. In this note, we study the spanning tree congestion of Hamming graphs.…”

“…For S ⊆ V(G), we denote the edge set of G[S ] by ι G (S ), and the boundary edge set by ) is known [6]. Also, stc(K d 2 ) is determined asymptotically [8].…”

Optimization of spanning tree stretch and congestion for graphs

“…林诒勋: 图的支撑树伸展与层叠最优化 (5) 象棋车图 K m × K n [32] , 证明 首先, 因为 P 2 ⊗ P 2 = K 4 是完全图, 由命题 2.3 得知 stc(P 2 ⊗ P 2 ) = 3. 其次, 对 P 3 ⊗ P 3 , 只有 4 个角的顶点 u、v、x 和 y 的度为 3, 其余顶点的度为 5 或 8.…”

“…例如, 已证明了对区间图、 分裂图、 置换图和凸二部图等具有 σ T (G) 3 (参见文献 [19,20,25,26]). 对特殊图的树层 stc(G) 有公式表示, 如完全二部图 K m,n 及平面格子图 P m × P n [27,28] 、完全 k-部图 K n1,n2,...,n k 及环面格子图 C m × C n [28,29] 、柱面格子图 C m × C n [30] 、三 角网格图 T n [31] 、象棋车图 K m × K n [32] 、外平面图 [33] 等. 关于图类刻画问题, 已知判定 σ T (G) 2 是多项式时间可解的, 而判定 σ T (G) k (k 4) 是 NP-完全的 (参见文献 [18,34]).…”