Improved Universal Graphs for Trees

TL;DR

The paper resolves, up to an improved constant, the Chung–Graham question on the minimal number of edges in an $n$-vertex graph universal for all $n$-vertex trees by constructing explicit universal graphs based on ternary trees. The core idea is a divide-and-conquer embedding into a ternary-tree-derived host, facilitated by refined separator lemmas and an eating-order framework that preserves admissibility during recursive embeddings. The main result tightens the upper bound to $s(n)\le\frac{19}{6\ln 3}\,n\ln n+O(n)$ and extends the approach to graphs of treewidth $w$, obtaining $nw\ln(n/w)-O(nw)\le s_w(n)\le\frac{19}{6\ln3}(w+1)n\ln(n/w)+O(nw)$. The technique relies on hierarchical graph-buildings $U_{n,d}$, precise edge-counting, and a blow-up argument to handle width-$w$ graphs, offering a robust path toward asymptotics for the tree-universal problem and its near-tree generalizations.

Abstract

A graph $G$ is universal for a class of graphs $\mathcal{C}$, if, up to isomorphism, $G$ contains every graph in $\mathcal{C}$ as a subgraph. In 1978, Chung and Graham asked for the minimal number $s(n)$ of edges in a graph with $n$ vertices that is universal for all trees with $n$ vertices. The currently best bounds assert that $n\ln n-O(n)\le s(n) \le C n\ln n+O(n)$, where $C = \frac{14}{5\ln 2} \approx 4.04$. Here, we improve the upper bound to $c n\ln n + O(n)$, where $c = \frac{19}{6\ln 3} \approx 2.88$. We develop in the proof a strategy that, broadly speaking, is based on separating trees into three parts, thus enabling us to embed them in a structure that originates from ternary trees. Our method also applies to graphs that are close to being trees, measured by their treewidth. Let $s_w(n)$ be the minimum number of edges in a $n$-vertex graph that is universal for graphs with treewidth $w$. By performing a graph blow-up to our universal structure and counting necessary edges carefully, we establish that $nw \ln(n/w) -O(nw) \leq s_w(n) \leq \frac{19}{6\ln3} n (w+1) \ln(n/w) + O(nw)$.

Figures (9)

• Figure 1: This is the ternary tree $T_{3,3}$ including the eating order. The vertex labels on level $\ell$ from $\{1,2,3\}^\ell$ are denoted within the circles, the position of the vertex in the eating order is denoted next to the circle. For example, the vertex with label $32$ is the $8$-th vertex to be eaten. Further, the arithmetic operations $32+2=11$ and $32-2=23$ on the vertices are indicated by the dashed arrows. The eating order on $V_{3,3}$ also determines the vertex sets of the $U_{n,3}$ of height $3$. The smallest such graph is $U_{14,3}$ with $V(U_{14,3})$ visualized by the dotted curve ($26$ vertices have been eaten) and the largest such graph is $U_{40,3}$ with $V(U_{40,3})=V_{3,3}$ (no vertex has been eaten).
• Figure 2: The admissible graph $A$ is given by its root $r_A$, the admissible subgraph $A'$ of $T^*_{h-1,d}$ and up to $d-1$ copies of $T^*_{h-1,d}$. The corresponding vertex sets and the graph structure are indicated in the figure. Moreover, $A$ is an induced subgraph of $T_{h,d}^*$ and thus contains edges given by the above construction.
• Figure 3: This figure illustrates the two possibilities of an admissible graph $A$ of type Figure \ref{['fig:fig-b']} and $h\geq 2$. Let $U$ be the induced subgraph containing the first $N+X_1+1$ vertices in the eating order of $A$. If $r_{A'}$ only has one child, $U$ is indicated by the dashed region in (a), otherwise in (b). By Observation \ref{['obs: admissible']}, $U$ is admissible in both cases. Therefore, we are able to use the induction hypothesis to embed any forest $F_1$ with $|F_1|\leq N+X_1$ into $U$ and thereby into $A$.
• Figure 4: This figure illustrates the three possible shapes of the graph in Figure \ref{['fig:IH1']} after the embedding of a forest $F_1$ with $X_1\leq |F_1|\leq N+X_1$. In case (c), we define $U$ as the induced subgraph containing the first $N+X_2+1$ vertices in the eating order. In the cases (a) and (b), we directly place the vertex $s$ at $r_{A'}$ indicated by the unfilled vertices, leaving a graph $U$. In any case, by Observation \ref{['obs: admissible']}, $U$ is admissible. Therefore, we are able to use the induction hypothesis to embed another forest $F_2$ with $|F_2|\leq X_2+1$ in (a), or $|F_2|\leq N+X_2+1$ in (b), or $|F_2|\leq N+X_2$ in (c), using $U$.
• Figure 5: This figure illustrates the three possibilities of an admissible graph $A$ with $h\geq 2$ of type Figure \ref{['fig:fig-c']} and restricted to the dotted region of type Figure \ref{['fig:fig-b']}. Let $U$ be the induced subgraph containing the first $2N+X_*+1$ vertices in the eating order of $A$. The graph $U$ is indicated by the dashed region divided according to the number of children of $r_{A'}$. By Observation \ref{['obs: admissible']}, $U$ is admissible in any case. Therefore, we are able to use the induction hypothesis to embed any forest $F_1$ with $|F_1|\leq 2N+X_*$ into $U$ and thereby into $A$.

Theorems & Definitions (23)

• Theorem 1
• Conjecture 2
• Theorem 3
• Lemma 4
• Lemma 5
• Definition 6
• Remark 7
• Lemma 9
• Lemma 10
• Corollary 11