The eccentricity of a vertex u in a graph G, denoted by epsilon G(u), is the maximum distance from u to other vertices in G. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity indices, denoted by sigma 0(G), sigma 1(G), and sigma 2(G), respectively. These are defined by sigma 0(G) = 1 u is an element of V(G) epsilon G(u), sigma 1(G) = & sum; and sigma 2(G) = & sum; |V (G)| u is an element of V(G) epsilon 2 G(u),uv is an element of E(G) epsilon G(u)epsilon G(v). We study lower and upper bounds on these parameters among n-vertex connected graphs with fixed diameter, chromatic number, clique number, or matching number. Most of the bounds are sharp, with the corresponding extremal graphs characterized. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of t-bar visibility representation of undirected graphs. A t-bar visibility representation of a digraph G assigns each vertex at most t horizontal bars in the plane so that there is an arc xy in the digraph if and only if some bar for x "sees" some bar for y above it along an unblocked vertical strip with positive width. The visibility number b(G) is the least t such that G has a t-bar visibility representation. In this paper, we solve several problems about b(G) posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The (n-l) $(n-\ell )$-deck of an n $n$-vertex graph is the multiset of subgraphs obtained from it by deleting l $\ell $ vertices. A family of n $n$-vertex graphs is l $\ell $-recognizable if every graph having the same (n-l) $(n-\ell )$-deck as a graph in the family is also in the family. We prove that the family of n $n$-vertex graphs with no cycles is l $\ell $-recognizable when n & GE;2l+1 $n\ge 2\ell +1$ (except for (n,l)=(5,2) $(n,\ell )=(5,2)$). As a consequence, the family of n $n$-vertex trees is l $\ell $-recognizable when n & GE;2l+1 $n\ge 2\ell +1$ and l & NOTEQUAL;2 $\ell \ne 2$. It is known that this fails when n=2l $n=2\ell $.

A graph is ℓ-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting ℓ vertices. We prove that strongly regular graphs with at least six vertices are 2-reconstructible.

A rooted tree is t-reconstructible if it is determined by its multiset of rooted subtrees (with the same root) obtained by deleting i vertices. We determine which rooted trees are - pound reconstructible for B <= 3 and show how this can be used to study reconstructibility of unrooted trees.

The (n − l)-deck of an n-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting l vertices. An n-vertex graph is l-reconstructible if it is determined by its (n − l)-deck, meaning that no other graph has the same deck. We prove that every tree with at least 6l+ 10 vertices is l-reconstructible.

A graph is l-reconstructible if it is determined by its multiset of subgraphs obtained by deleting l verties. Using centroids and rooted trees, we prove that trees with at least 22 vertices are 3-reconstructible.

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that $3$-regular graphs are $2$-reconstructible.

The (n − l)-deck of an n-vertex graph is the multiset of subgraphs obtained from it by deleting l vertices. A family of n-vertex graphs is l-recognizable if every graph having the same (n − l)-deck as a graph in the family is also in the family. We prove that the family of n-vertex graphs having no cycles is l-recognizable when n ≥ 2l+ 1 (except for (n, l) = (5, 2)). It is known that this fails when n = 2l.

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. For the complete bipartite graph $K_{m,n}$, the lower bound $b(K_{m,n})\ge\lceil{\frac{mn+4}{2m+2n}}\rceil$ from Euler's Formula is well known. We prove that equality holds.

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.

A t-bar visibility representation of a graph G assigns each vertex up to t horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least t such that G has a t-bar visibility representation is the bar visibility number of G, denoted by b(G). We show that if H is a spanning subgraph of G, then b(H)≤b(G)+1. It follows that b(G)≤⌈n∕6⌉+1 when G is an n-vertex graph. This improves the upper bound obtained by Chang et al. (2004).

Proposed problems should be submitted online atamericanmathematicalmonthly.submittable.com/submit.Proposed solutions to the problems below should be submitted by October 31, 2020, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.

We give a new proof of the theorem of Boesch-Tindell and Farzad-Mahdian-Mahmoodian-Saberi-Sadri that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. Our proof bounds the number of edges needed for such an extension; we give examples to demonstrate sharpness. We apply the characterization to a problem on non-transitive dice.

The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list of an $n$-vertex graph is $3$-reconstructible when $n\ge7$, and the threshold on $n$ is sharp. Using this result, we show that when $n\ge7$ the $(n-3)$-deck also determines whether an $n$-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are $2$-reconstructible when $n\ge6$, which are also sharp.

For a graph G, let $$f_2(G)$$ denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of $$f_2(G)$$ over 3-regular n-vertex simple graphs G. To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most $$\max \{0,\lfloor (c-1)/2\rfloor \}$$ vertices. More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most $$\max \{0,\lfloor (3n-2m+c-1)/2\rfloor \}$$ vertices. These bounds are sharp; we describe the extremal multigraphs.

Let F be a family of graphs. A graph G is F-saturated if G contains no member of F as a subgraph but G+e contains some member of F whenever e∈E(G¯). The saturation number and extremal number of F, denoted sat(n,F) and ex(n,F) respectively, are the minimum and maximum numbers of edges among n-vertex F-saturated graphs. For k∈N, let Fk and Fk′ be the families of k-connected and k-edge-connected graphs, respectively. Wenger proved sat(n,Fk)=(k−1)n−k2; we prove sat(n,Fk′)=(k−1)(n−1)−nk+1k−12. We also prove ex(n,Fk′)=(k−1)n−k2 and characterize when equality holds. Finally, we give a lower bound on the spectral radius for Fk-saturated and Fk′-saturated graphs.