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Infinitely many trees with maximum number of holes zero, one, and two. (English) Zbl 1437.05075
Summary: An $$L(2,1)$$-coloring of a simple connected graph $$G$$ is an assignment $$f$$ of nonnegative integers to the vertices of $$G$$ such that $$|f(u)-f(v)| \geqslant 2$$ if $$d(u,v)=1$$ and $$|f(u)-f(v)|\geqslant 1$$ if $$d(u,v)=2$$ for all $$u,v\in V(G)$$, where $$d(u,v)$$ denotes the distance between $$u$$ and $$v$$ in $$G$$. The span of $$f$$ is the maximum color assigned by $$f$$. The span of a graph $$G$$, denoted by $$\lambda (G)$$, is the minimum of span over all $$L(2,1)$$-colorings on $$G$$. An $$L(2,1)$$-coloring of $$G$$ with span $$\lambda(G)$$ is called a span coloring of $$G$$. An $$L(2,1)$$-coloring $$f$$ is said to be irreducible if there exists no $$L(2,1)$$-coloring g such that $$g(u) \leqslant f(u)$$ for all $$u \in V(G)$$ and $$g(v) < f(v)$$ for some $$v \in V(G)$$. If $$f$$ is an $$L(2,1)$$-coloring with span $$k$$, then $$h \in\{0,1, 2, \dots, k\}$$ is a hole if there is no $$v \in V(G)$$ such that $$f(v)=h$$. The maximum number of holes over all irreducible span colorings of $$G$$ is denoted by $$H_\lambda (G)$$. A tree $$T$$ with maximum degree $$\Delta$$ having span $$\Delta +1$$ is referred to as Type-I tree; otherwise it is Type-II. In this paper, we give a method to construct infinitely many trees with at least one hole from a one-hole tree and infinitely many two-hole trees from a two-hole tree. Also, using the method, we construct infinitely many Type-II trees with maximum number of holes one and two. Further, we give a sufficient condition for a Type-II tree with maximum number of holes zero.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C05 Trees
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##### References:
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