Academic literature on the topic 'Graceful tree conjecture'

A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma and Hoede that there is an equivalent conjecture for GTC stating that all trees containing a perfect matching are strongly graceful (graceful with an extra condition). In this paper we extend the above result by showing that there exist infinitely many equivalent versions of the GTC. Moreover we verify these infinitely many equivalent conjectures of GTC for trees of diameter at most 7. Among others we are also able to identify new graceful trees and in particular generalize the ?-construction of Stanton-Zarnke (and later Koh- Rogers-Tan) for building graceful trees through two smaller given graceful trees.

The security of passwords generated by the graphic lattices is based on the difficulty of the graph isomorphism, graceful tree conjecture, and total coloring conjecture. A graphic lattice is generated by a graphic base and graphical operations, where a graphic base is a group of disjointed, connected graphs holding linearly independent properties. We study the existence of graphic bases with odd-graceful total colorings and show graphic lattices by vertex-overlapping and edge-joining operations; we prove that these graphic lattices are closed to the odd-graceful total coloring.

A variety of labelings on trees have emerged in order to attack the Graceful Tree Conjecture, but lack showing the connections between two labelings. In this paper, we propose two new labelings: vertex image-labeling and edge image-labeling, and combine new labelings to form matching-type image-labeling with multiple restrictions. The research starts from the set-ordered graceful labeling of the trees, and we give several generation methods and relationships for well-known labelings and two new labelings on trees.

<p>A tree <span class="math"><em>T</em>(<em>V</em>, <em>E</em>)</span> is <span><em>graceful</em></span> if there exists an injective function <span class="math"><em>f</em></span> from the vertex set <span class="math"><em>V</em>(<em>T</em>)</span> into the set <span class="math">{0, 1, 2, ..., ∣<em>V</em>∣ − 1}</span> which induces a bijective function <span class="math"><em>f</em>ʹ</span> from the edge set <span class="math"><em>E</em>(<em>T</em>)</span> onto the set <span class="math">{1, 2, ..., ∣<em>E</em>∣}</span>, with <span class="math"><em>f</em>ʹ(<em>u</em><em>v</em>) = ∣<em>f</em>(<em>u</em>) − <em>f</em>(<em>v</em>)∣</span> for every edge <span class="math">{<em>u</em>, <em>v</em>} ∈ <em>E</em></span>. Motivated by the conjecture of Alexander Rosa (see) saying that all trees are graceful, a lot of works have addressed gracefulness of some trees. In this paper we show that some uniform trees are graceful. This results will extend the list of graceful trees.</p>

One of the most famous open problems in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In this paper, we define graceful labelings for countably infinite graphs, and state and verify a Graceful Tree Conjecture for countably infinite trees.

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that, when each edge is assigned the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study these problems in the context of graph games. The graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to m. Alice’s goal is to gracefully label the graph as Bob’s goal is to prevent it from happening. In this work, we present the first results in this area by showing winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths.

That he marks the fall of every sparrow was duly considered by Brother Clothren as he keyed the designation "Denumerable" into the blank spaces following the prompt CARDINALITY? blinking in pale violet on his workscope. A fitting designation, he thought for the paternal man-god of an insignificant water planet-- a god beset on one hand by the most ordinary of human emotions and on the other serving as the crux of impenetrable mysteries. By any measure though, this Yahweh was a small figure indeed compared to deities of higher cardinality such as the ineffably beautiful Aleph Three god of the aboreals of Ka'a Ntg whose very name can only be approximated by a metaphorical conjunction of the galactagrams for "starwheel", "phoenix", and "beyond all attributes". For countless millennia the graceful metallocrete arch of Brother Clothren's Order of Remembrance has lifted its motto in archaic Fa'an into the dazzling night sky of Kletha. Etched on crystal lithium, its sinuous script reads, "Choose as gods might choose, those few things to be honored with perpetual remembrance." The surprising proof of Meinong's Conjecture in ages past had established the existence of non-intersecting realms of deities as irrefutable fact - deities wrought into existence solely by the collective belief and adoration of the worshipful races. Although Brother Clothrens's ancient order originally strove to record the detailed history of each such race, such aspirations proved too daunting owing to their unexpected large numbers. As a result, an early Abbot -- the stately Haa'tan - reluctantly issued a Standing Order limiting historical accounts to the quintessence of each race -- namely, a detailed account of its gods. In this, the Abbott was surely right, thought Brother Clothren recalling one of Haa'tan's more popular quotes, "Tell me about your gods and I will reveal the inmost secrets of your heart". Nevertheless, even Brother Clothren in his chosen vocation of Archeotheist, sometimes murmured at preserving such a minuscule part of the vast and glorious history of vanished races while consigning the large measure of their past to certain oblivion. Nevertheless, like other members of his ancient Order, each night the meek and obedient Brother Clothren dreamed the eventless white dream of the pure. For the account of the Terrene god a final page still remained. Brother Clothren continued keying in text, "...that such a simple contingency could spell the demise of a deity of his cardinality was not only inconceivable to his devotees but also beyond the ken of the very deity who would himself succumb as its victim..." Brother Clothren paused for a moment contemplating the deep mystery of divine existences then continued,. . ."On April 7, 1984 (by local solar reckoning) in Estill Fork, Alabama, a solitary sparrow alighted erratically on a redbud tree in Otis Farnborough's side yard. Underneath, a strutting rooster in vivid metallic hues imperiously herded three clucking hens scratching for grubs. The sparrow had flown south from Tennessee into the mountain fastness of the upper Paint Rock Valley in North Alabama where it had contracted aviomycosis and was terminally ill. There on Otis Farnborough's redbud tree its long journey ended as its eyes glazed over, its body spasmed, and then, releasing its grip, the dead bird toppled headlong into a clutch of startled hens. At that precise moment, the ceaseless purview of the ever-watchful Yahweh was exceeded and he became zero in all his parts signifying his total extinction."

碩士
東海大學
應用數學系
101
The well known Graceful Tree Conjecture(GTC) claimed that all trees are graceful, which still remains open until today. It was proved in 1999 by H. Broersma and C. Hoede that there is an equivalent conjecture for GTC that all trees containing a perfect matching is strongly graceful. In this thesis we verify by extending the above result that there exist infinitely many equivalent versions of the GTC. More precisely, for a fixed graceful tree Tk of order k, we show that for each k ≥ 2, the conjecture that all trees containing a graceful Tk-factor is strongly Tk-graceful is equivalent to the conjecture that all trees are graceful. More applications are also included by way of identifying new classes of graceful graphs. In particular we verify infinitely many equivalent Tk-version conjectures of GTC for those trees of diameter no more than 2⌈D(Tk)2⌉ + 5, where D(Tk) is the diameter of Tk.

This chapter considers problems of whether a graph can be decomposed into certain other kinds of graphs, primarily cycles. It begins with a background on nineteenth-century mathematician Thomas Penyngton Kirkman and the problem he invented known as Kirkman's Schoolgirl Problem, stated as: How many triples can be formed with x symbols in such a way that no pair of symbols occurs more than once in the triple? This is followed by a discussion of the Steiner triple system, the relationship between cyclic decomposition problems and a problem called Alspach's Conjecture, graceful graphs, and the Graceful Tree Conjecture. The chapter concludes with an analysis of the puzzle dubbed Instant Insanity and how graphs can be utilized to solve it.

This study investigated how students in Grades 4, 6, and 8 reasoned through a non-routine, unsolved problem. The study took place at a K-8 school in the Midwestern United States. Each grade participated in two or three task-based sessions lasting between 45 and 60 minutes with the researchers. During the sessions, students engaged in the Graceful Tree Conjecture where they examined graceful labelling for Star, Path, and Caterpillar Graphs. We examined differences in students’ generalized solutions across the grades and how they were able to provide justifications and state generalizations of a graceful labelling for the graphs in the Path Class. Descriptions of students’ generalized solutions are included for each grade level.

A graceful labelling of a tree T is an injective function f: V (T) → {0, 1, . . . , |E(T)|} such that {|f(u)−f(v)|: uv ∈ E(T)} = {1, 2, . . . , |E(T)|}. A tree T is said to be 0-rotatable if, for any v ∈ V (T), there exists a graceful labelling f of T such that f(v) = 0. In this work, it is proved that the follow- ing families of caterpillars are 0-rotatable: caterpillars with perfect matching; caterpillars obtained by identifying a central vertex of a path Pn with a vertex of K2; caterpillars obtained by identifying one leaf of the star K1,s−1 to a leaf of Pn, with n ≥ 4 and s ≥ ⌈n−1 2 ⌉; caterpillars with diameter five or six; and some families of caterpillars with diameter at least seven. This result reinforces the conjecture that all caterpillars with diameter at least five are 0-rotatable.