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• W1524617299 abstract "Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)rightarrowmathbb{Z}_{ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is graceful if the labels of the vertices are ${0, 1, ldots , |V(G)| - 1}$, and the induced labels of the edges are distinct. The graph $G$ is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful labeling. In this paper, we prove that if $T$ is a tree with central vertex and root $v$, such that each vertex not in the last two levels has an odd number of children, and $T$ satisfies one of the following conditions (a)-(e), then $T$ has a graceful labeling $f$ with $f(v) = 0$: (a) $T$ is a diameter-6 complete tree; (b) $T$ is a diameter-6 tree such that no two leaves of distance 2 from $v$ are siblings, and each leaf of distance 2 from $v$ has a sibling with an even number of children; (c) $T$ is a diameter-$2r$ complete tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3pmod{4}$; (d) $T$ is a diameter-$2r$ tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3pmod{4}$, no two leaves of distance $r - 1$ from $v$ are siblings, and each leaf of distance $r - 1$ from $v$ has a sibling with an even number of children; (e) $T$ is a diameter-6 tree, such that each internal vertex has an odd number of children. In particular, all depth-3 trees of which each internal vertex has an odd number of children are graceful." @default.
• W1524617299 created "2016-06-24" @default.
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• W1524617299 date "2014-02-22" @default.
• W1524617299 modified "2023-10-01" @default.
• W1524617299 title "New graceful diameter-6 trees by transfers" @default.
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