On maximum graphs in Tutte polynomial posets

TL;DR

This work shows that the uniformly optimally reliable graphs identified for edge-sparse classes are in fact Tutte-maximum for a wide range of invariants derived from the Tutte polynomial $T(G;x,y)$. Using the $(n,m)$ Tutte polynomial poset framework, the authors prove that in $\mathcal{G}_{n,n}$ the maximum is the cycle $C_n$, in $\mathcal{G}_{n,n+1}$ the maximum is the most-even theta graph, and in $\mathcal{G}_{n,n+2}$ the maximum is a box graph $B_n^*$ with ears as evenly distributed as possible. These results imply that the maximum graphs maximize not only all-terminal reliability but also coefficients and generating functions related to spanning trees, forests, and various orientations, among other Tutte-derived invariants. The paper thereby unifies several known UOR results under the Tutte poset perspective and outlines conjectures for higher-density classes, highlighting both possible maxima and the potential for non-uniqueness in some cases. This framework provides a systematic method to identify Tutte-maximum graphs across broad families and invariants.

Abstract

Boesch, Li, and Suffel were the first to identify the existence of uniformly optimally reliable graphs (UOR graphs), graphs which maximize all-terminal reliability over all graphs with $n$ vertices and $m$ edges. The all-terminal reliability of a graph, and more generally a graph's all-terminal reliability polynomial $R(G;p)$, may both be obtained via the Tutte polynomial $T(G;x,y)$ of the graph $G$. Here we show that the UOR graphs found earlier are in fact maximum graphs for the Tutte polynomial itself, in the sense that they are maximum not just for all-terminal reliability but for a vast array of other parameters and polynomials that may be obtained from $T(G;x,y)$ as well. These parameters include, but are not limited to, enumerations of a wide variety of well-known orientations, partial orientations, and fourientations of $G$; the magnitudes of the coefficients of the chromatic and flow polynomials of $G$; and a wide variety of generating functions, such as generating functions enumerating spanning forests and spanning connected subgraphs of $G$. The maximality of all of these parameters is done in a unified way through the use of $(n,m)$ Tutte polynomial posets.

Figures (4)

• Figure 1: The two maximal graphs of the $(7,11)$ Tutte polynomial poset. The left graph maximizes all-terminal reliability, and the right graph maximizes acyclic orientations, in the $\mathcal{G}_{7,11}$ graph class. Both parameters may be obtained from the Tutte polynomial.
• Figure 2: An illustration of the graphs $G$ (left) and $H$ (right) from Theorem \ref{['comb']}. Additional blocks, if present, are identical in the two graphs and are not shown.
• Figure 3: Delta graphs $\Delta(a,b,c,d,e)$, box graphs $B(a,b,c,d,e,f)$, and cylinder graphs $C(a,b,c,d,e,f)$. Each labeled line indicates a path with length in edges equal to the label.
• Figure 4: The smallest non-chain Tutte polynomial poset, the $(6,7)$ Tutte polynomial poset. (If multiple graphs have the same Tutte polynomial only one graph is shown.) The graph $\theta(2,2,2) \cdot K_2$ is incomparable with both the graphs $\theta(1,2,4)$ and $C_3 \cdot C_4$. As one moves upward in the poset, a wide variety of graph parameters associated with the Tutte polynomial increase.

Theorems & Definitions (27)

• Theorem 1.1: k22
• Definition 1.2
• Theorem 2.1: hpr10
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• Corollary 3.3
• proof
• Lemma 4.1