有向图上的交错伴随
Interleaved adjoints on directed graphs
摘要：
对于整数k >= 1，有向图G的第k个交织伴随是具有顶点集V（G）^k和弧（（u_1，.，u_k），（v_1，.，v_k））使得（u_i，v_i）\in A（G）对于i = 1，.，k和（v_i，u_{i+1}）\in A（G），其中i = 1，.，k-1对于任意k，我们得到了i_k（G）的色数关于G的色数的上下界.特别地，我们发现传递竞赛图的交错伴随的色数的紧界。我们将这个结果与伴随函子的范畴性质结合起来，得到以下结果。对于任意整数el，存在代数长度为el的有向路Q_{\ell}，它允许同态进入任意色数至少为4的有向图.我们讨论了这种方法的多因素版本的弱Hedetniemi猜想可能产生的影响。
英文摘要：
For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, ..., u_k), (v_1, ..., v_k)) such that (u_i,v_i) \in A(G) for i = 1, ..., k and (v_i, u_{i+1}) \in A(G) for i = 1, ..., k-1. For every k we derive upper and lower bounds for the chromatic number of i_k(G) in terms of that of G. In particular, we find tight bounds on the chromatic number of interlacing adjoints of transitive tournaments. We use this result in conjunction with categorial properties of adjoint functors to derive the following consequence. For every integer ell, there exists a directed path Q_{\ell} of algebraic length ell which admits homomorphisms into every directed graph of chromatic number at least 4. We discuss a possible impact of this approach on the multifactor version of the weak Hedetniemi conjecture.
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