منابع مشابه
Digraph Girth via Chromatic Number
Let D be a digraph. The chromatic number χ(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or ∞ if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with χ(D) > k. It is shown that G(k, n) ≥ n1/k and G(k, n) ≤ (3 log2 n...
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This paper will look at the relationship between high girth and high chromatic number in both its finite and transfinite incarnations. On the one hand, we will demonstrate that it is possible to construct graphs with high oddgirth and high chromatic number in all cases. We will then look at a theorem which tells us why, at least in the transfinite case, it is impossible to generalize this to in...
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متن کاملThe distinguishing chromatic number of bipartite graphs of girth at least six
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1978
ISSN: 0012-365X
DOI: 10.1016/0012-365x(78)90102-4