Measures of closeness to cordiality for graphs

TL;DR

This paper introduces two quantitative measures, ${\mathscr D}_1(G)$ and ${\mathscr D}_2(G)$, to capture how far a graph is from admitting a cordial labelling. It first establishes general join bounds: ${\mathscr D}_1(G_1+G_2) \le {\mathscr D}_1(G_1){\mathscr D}_1(G_2) + {\mathscr D}_1(G_1) + {\mathscr D}_1(G_2)$ and ${\mathscr D}_2(G_1+G_2) \le {\mathscr D}_2(G_1) + {\mathscr D}_2(G_2) + 1$, then computes exact values or tight bounds for a suite of graph families including ${\mathcal T}_n$, ${\mathcal K}_n$, ${\mathcal K}_{n_1,\ldots,n_r}$, ${\mathcal C}_n$, ${\mathcal W}_n$, and ${\mathcal F}_{m,n}$. Key findings include precise formulas for ${\mathscr D}_1$ and ${\mathscr D}_2$ on trees, complete graphs, and complete multipartite graphs, as well as parity-driven results for cycles, wheels, and fans; these results also yield a cordiality criterion for complete multipartite graphs (at most three odd part sizes). The work provides a unified framework to quantify proximity to cordiality and poses open problems about exact values in multipartites and extremal questions for these measures. The methods combine constructive labellings, parity analysis, and join-graph decompositions to derive both exact values and sharp bounds.

Abstract

A graph $G$ is cordial if there exists a function $f$ from the vertices of $G$ to $\{0,1\}$ such that the number of vertices labelled $0$ and the number of vertices labelled $1$ differ by at most $1$, and if we assign to each edge $xy$ the label $|f(x)-f(y)|$, the number of edges labelled $0$ and the number of edges labelled $1$ also differ at most by $1$. We introduce two measures of how close a graph is to being cordial, and compute these measures for a variety of classes of graphs.

Theorems & Definitions (9)

• Theorem 2.1
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• Corollary 5.2
• Theorem 6.1
• Corollary 6.2
• Theorem 7.1
• Theorem 8.1