Subtopic Deep Dive
Antimagic Labeling of Graphs
Research Guide
What is Antimagic Labeling of Graphs?
Antimagic labeling of a graph assigns distinct integers from 1 to |E| to the edges such that the sums of labels incident to each vertex are all distinct.
This labeling ensures pairwise distinct vertex sums using consecutive integers. Research focuses on existence for dense graphs, regular graphs, and specific families like cycles and complete graphs. Over 20 papers since 2003, with Alon et al. (2003, 105 citations) proving dense graphs are antimagic.
Why It Matters
Antimagic labelings distinguish vertices via edge label sums, advancing bijective graph labelings. Alon et al. (2004, 102 citations) showed applications to dense graphs with minimum degree Ω(n). Cranston (2008, 58 citations) proved regular bipartite graphs antimagic, impacting combinatorial designs. Chang et al. (2015, 82 citations) addressed regular graphs, enabling constructions in network coding and cryptography protocols.
Key Research Challenges
Antimagic for Sparse Graphs
Proving antimagic labelings exist for graphs with low degrees remains open beyond Hartsfield-Ringel conjecture cases. Alon et al. (2003, 105 citations) handled dense graphs, but trees and sparse graphs lack general results. Hefetz (2005, 54 citations) used Combinatorial Nullstellensatz for specific cases.
Nonbipartite Even Regular Graphs
Determining antimagic status for even-degree nonbipartite regular graphs is unresolved. Chang et al. (2015, 82 citations) proved odd-degree cases, noting bipartite even regular are antimagic. Cranston et al. (2014, 59 citations) advanced odd-degree regulars.
Vertex vs Edge Antimagic
Extending edge-antimagic to vertex-antimagic colorings poses distinct sum challenges. Arumugam et al. (2017, 127 citations) introduced local antimagic vertex coloring. Bača and Miller (2008, 66 citations) surveyed super edge-antimagic problems.
Essential Papers
Local Antimagic Vertex Coloring of a Graph
S. Arumugam, K. Premalatha, Martin Bača et al. · 2017 · Graphs and Combinatorics · 127 citations
Dense graphs are antimagic
Noga Alon, Gil Kaplan, Arieh Lev et al. · 2003 · arXiv (Cornell University) · 105 citations
An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a ve...
Antimagic Labeling of Regular Graphs
Fei-Huang Chang, Yu‐Chang Liang, Zhishi Pan et al. · 2015 · Journal of Graph Theory · 82 citations
A graph is antimagic if there is a one-to-one correspondence such that for any two vertices , . It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree...
Super Edge-Antimagic Graphs: A Wealth of Problems and Some Solutions
Martin Bača, Mirka Miller · 2008 · 66 citations
Graph theory, and graph labeling in particular, are fast-growing research areas in mathematics. New results are constantly being discovered and published at a rapidly increasing rate due to the eno...
Regular Graphs of Odd Degree Are Antimagic
Daniel W. Cranston, Yu‐Chang Liang, Xuding Zhu · 2014 · Journal of Graph Theory · 59 citations
Abstract An antimagic labeling of a graph G with m edges is a bijection from to such that for all vertices u and v , the sum of labels on edges incident to u differs from that for edges incident to...
Regular bipartite graphs are antimagic
Daniel W. Cranston · 2008 · Journal of Graph Theory · 58 citations
Abstract A labeling of a graph G is a bijection from E ( G ) to the set {1, 2,… | E ( G )|}. A labeling is antimagic if for any distinct vertices u and v , the sum of the labels on edges incident t...
Anti‐magic graphs via the Combinatorial NullStellenSatz
Dan Hefetz · 2005 · Journal of Graph Theory · 54 citations
Abstract An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,…, m such that all n vertex sums are pairwise distinct, where a vertex s...
Reading Guide
Foundational Papers
Start with Alon et al. (2003, 105 citations) for dense graphs proof and definition; Cranston (2008, 58 citations) for bipartite regulars; Bača and Miller (2008, 66 citations) for edge-antimagic survey.
Recent Advances
Arumugam et al. (2017, 127 citations) on local antimagic vertex coloring; Chang et al. (2015, 82 citations) on regular graphs; Cranston et al. (2014, 59 citations) on odd-degree regulars.
Core Methods
Probabilistic existence (Alon et al., 2004); algebraic Combinatorial Nullstellensatz (Hefetz, 2005); degree-based constructions (Cranston et al., 2014; Chang et al., 2015).
How PapersFlow Helps You Research Antimagic Labeling of Graphs
Discover & Search
Research Agent uses searchPapers and citationGraph to map 250M+ papers, starting from Alon et al. (2003) to find descendants like Cranston (2008). exaSearch queries 'antimagic labeling regular graphs' for 50+ results; findSimilarPapers expands from Chang et al. (2015) to related regular graph proofs.
Analyze & Verify
Analysis Agent applies readPaperContent to extract proofs from Alon et al. (2004), then verifyResponse with CoVe checks conjecture claims against citations. runPythonAnalysis simulates vertex sums on graph examples from Cranston et al. (2014) using NetworkX, with GRADE scoring evidence strength for Hartsfield-Ringel conjecture.
Synthesize & Write
Synthesis Agent detects gaps like even-degree nonbipartite regulars via contradiction flagging across Bača and Miller (2008). Writing Agent uses latexEditText, latexSyncCitations for proofs, latexCompile for manuscripts, and exportMermaid diagrams antimagic spectra flows.
Use Cases
"Verify if C_5 cycle graph has antimagic labeling using Python simulation"
Research Agent → searchPapers('antimagic cycles') → Analysis Agent → runPythonAnalysis(NetworkX cycle graph, label 1-5 edges, compute vertex sums distinctness) → researcher gets plot of sums and boolean verification.
"Draft LaTeX proof extending antimagic to new graph family"
Synthesis Agent → gap detection on Cranston (2008) → Writing Agent → latexEditText(proof skeleton) → latexSyncCitations(Alon 2003 et al.) → latexCompile → researcher gets compiled PDF with figure.
"Find GitHub code for antimagic labeling algorithms"
Research Agent → citationGraph(Chang 2015) → Code Discovery (paperExtractUrls → paperFindGithubRepo → githubRepoInspect) → researcher gets repo links with verified labeling scripts.
Automated Workflows
Deep Research workflow scans 50+ antimagic papers via searchPapers → citationGraph, producing structured report with Hartsfield-Ringel progress table. DeepScan applies 7-step CoVe to verify claims in Hefetz (2005) Combinatorial Nullstellensatz proofs. Theorizer generates conjectures for even regular graphs from Cranston et al. (2014) patterns.
Frequently Asked Questions
What is an antimagic labeling?
A bijection from edges to {1, ..., |E|} such that vertex sums of incident edge labels are pairwise distinct (Alon et al., 2003).
What are key methods in antimagic labeling?
Probabilistic methods for dense graphs (Alon et al., 2004); Combinatorial Nullstellensatz for specific graphs (Hefetz, 2005); constructive labelings for regulars (Cranston, 2008; Chang et al., 2015).
What are foundational papers?
Alon et al. (2003, 105 citations) proves dense graphs antimagic; Cranston (2008, 58 citations) for bipartite regulars; Bača and Miller (2008, 66 citations) surveys edge-antimagic.
What open problems exist?
Hartsfield-Ringel conjecture: all connected graphs ≥3 vertices antimagic; even-degree nonbipartite regulars (Chang et al., 2015); full antimagic spectra for cycles.
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