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Coin Graph Recognition in Complete Trees Research

Explore the relationship between coin graphs and complete trees, focusing on fitting coins on leaf nodes in tree structures. Discover insights on tree arity versus space requirements for leaf nodes and the maximum height of various n-ary trees. Learn about the findings distinguishing complete trees from coin graphs and the implications for tree structures.

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Coin Graph Recognition in Complete Trees Research

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  1. Coin-Graph Recognition in Complete Trees • Amit Kumar Dey • AftabHussain • AnnajiatAlimRasel • DipankarChaki Joy

  2. Considering Trees

  3. Fitting coins tree in a circle running over leaf nodes • 1 edge = 2r • R = h x 2r • Circumference =2πR = 2π(h x 2r) = 4 π rh • Number of nodes at leaf = 2h • Total Diameter of nodes =2h x 2r = 2h+1 x r r r r r r r

  4. Tree arity Vs space required for leaf nodes Tree Height • combined diameter of leaf nodes

  5. Tree arity Vs space required for leaf nodes Tree Height • 100 base log of combined diameter of leaf nodes

  6. Research Outcome • Today’s findings: • Maximum height of tree • Binary tree: 5 • Ternary tree: 2 • Quarternary tree: 1 • 5-ary tree: 1 • 6-ary tree: 0 (It can be shown from the work done in 1st brainstorming workshop) • A complete tree is NOT a coin graph (sufficient condition) • it is n-ary tree (n>=6) • Its height is longer than mentioned above

  7. 1st workshop Findings: Tree is not a coin graph if there exists any vertex with degree > 5 • So, 6-ary tree is NOT possible • (becomes a wheel graph with a cycle)

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