[Paper Review] Connectedness and Hamiltonicity of graphs on vertex colorings

Daniel C. McDonald|arXiv (Cornell University)|Jul 19, 2015

Graph Labeling and Dimension Problems8 references3 citations

TL;DR

This paper introduces and analyzes the graphs $ G^j_k(H) $, where vertices represent proper $ k $-colorings of a graph $ H $, and edges connect colorings differing on a connected subgraph of at most $ j $ vertices. The key contribution is establishing that $ h_{k+1}(H) = 1 $ if and only if each partite set of $ H $ has odd size, proving a necessary and sufficient condition for Hamiltonicity in $ G^1_{k+1}(H) $ when $ H $ is a complete $ k $-partite graph with odd-sized parts.

ABSTRACT

Given a graph $H$, let $G^j_k(H)$ be the graph whose vertices are the proper $k$-colorings of $H$, with edges joining two colorings if $H$ contains a connected subgraph on at most $j$ vertices that includes all vertices where the colorings differ. Properties of $G^1_k(H)$ have been investigated before, including connectedness and Hamiltonicity. We introduce and study the parameters $g_k(H)$ and $h_k(H)$, which denote the minimum $j$ such that $G^j_k(H)$ is connected or Hamiltonian, respectively.

Motivation & Objective

• To define and analyze the graph $ G^j_k(H) $, whose vertices are proper $ k $-colorings of a graph $ H $, with edges connecting colorings that differ on a connected subgraph of size at most $ j $.
• To introduce and study the parameters $ g_k(H) $ and $ h_k(H) $, the minimal $ j $ such that $ G^j_k(H) $ is connected or Hamiltonian, respectively.
• To determine necessary and sufficient conditions for $ h_k(H) = 1 $, particularly in the context of complete $ k $-partite graphs with odd-sized partite sets.
• To establish a structural characterization linking the parity of partite set sizes to the Hamiltonicity of the coloring graph $ G^1_k(H) $.

Proposed method

• Define $ G^j_k(H) $ as a graph with proper $ k $-colorings of $ H $ as vertices, and an edge between two colorings if they differ on a connected subgraph of at most $ j $ vertices.
• Use the mixing process on vertex colorings to explore transitions between proper colorings via single-vertex recoloring steps.
• Construct Hamiltonian cycles in $ G^1_{k+1}(H) $ by combining paths in hypercubes $ Q_{|X_i|}(b_i, c_i) $ that traverse all colorings using two colors on each partite set $ X_i $, excluding monochromatic extremes.
• Prove that a Hamiltonian cycle in $ G^1_{k+1}(H) $ exists only if each partite set $ X_i $ has odd size, by showing such a cycle induces a Hamiltonian path between $ b\cdots b $ and $ c\cdots c $ in the hypercube $ Q_{|X_i|}(b,c) $.
• Use induction and path concatenation across partite sets to build a full Hamiltonian cycle through $ G^1_{k+1}(H) $, ensuring all proper $ (k+1) $-colorings are included exactly once.
• Establish that if $ h_{k+1}(H) = 1 $, then each $ |X_i| $ must be odd, by analyzing the structure of adjacent colorings and the constraints on transitions in $ G^1_{k+1}(H) $.

Experimental results

Research questions

• RQ1 What is the minimal $ j $ such that $ G^j_k(H) $ is connected for a given graph $ H $ and integer $ k $?
• RQ2 What is the minimal $ j $ such that $ G^j_k(H) $ is Hamiltonian for a given graph $ H $ and integer $ k $?
• RQ3 Under what conditions is $ G^1_k(H) $ Hamiltonian when $ H $ is a complete $ k $-partite graph?
• RQ4 Can the parity of partite set sizes in a complete $ k $-partite graph determine the Hamiltonicity of $ G^1_k(H) $?
• RQ5 Is $ h_k(H) = 1 $ if and only if all partite sets in $ H $ have odd size?

Key findings

• The parameter $ h_{k+1}(H) = 1 $ if and only if every partite set $ X_i $ in $ H $ has an odd number of vertices, when $ H $ is a complete $ k $-partite graph.
• A Hamiltonian cycle exists in $ G^1_{k+1}(H) $ when each $ |X_i| $ is odd, constructed by concatenating paths in hypercubes $ Q_{|X_i|}(b_i, c_i) $ that exclude the monochromatic vertices $ b_i\cdots b_i $ and $ c_i\cdots c_i $.
• If $ h_{k+1}(H) = 1 $, then each $ |X_i| $ must be odd, as otherwise no Hamiltonian path can exist between $ b\cdots b $ and $ c\cdots c $ in the corresponding hypercube.
• The construction of the Hamiltonian cycle in $ G^1_{k+1}(H) $ ensures every proper $ (k+1) $-coloring of $ H $ is included exactly once, covering both colorings using only $ k $ colors and those using all $ k+1 $ colors.
• The adjacency structure in $ G^1_{k+1}(H) $ ensures that transitions between paths in different partite sets are valid, relying on the non-adjacency of indices $ a_i $ and $ a_{i+1} $ in the cycle.