This document contains a list of open problems that Duc A. Hoang is interested in. It is also available in PDF format. Please get in touch if you have any idea/comment/question/update. You can find (probably more interesting) open problems from the following resources.
Graph Theory - Favorite Conjectures and Open Problems - 1 (edited by Ralucca Gera, Stephen Hedetniemi, Craig Larson) and Graph Theory - Favorite Conjectures and Open Problems - 2 (edited by Ralucca Gera, Teresa W. Haynes, Stephen T. Hedetniemi).
Unless otherwise noted, all graphs are simple, undirected. We often denote by the path, cycle, complete graph, complete bipartite graph on vertices, respectively. For two graphs , the disjoint union of and , denoted by , is the graph with and . For the terminology and notation not defined here, see the textbooks by Diestel  or West . For more details on Combinatorial Reconfiguration (https://en.wikipedia.org/wiki/Reconfiguration), see the surveys [13, 12, 11, 5] and the wiki page https://reconf.wikidot.com/.
The double-broom graph is the tree obtained from a path by attaching leaves to and leaves to , where are positive integers. Two independent sets are adjacent under Token Sliding if there exists such that , , and . The Token Sliding problem asks whether there is a -sequence between and in , that is, a sequence of independent sets starting with and ending with where any two consecutive members are adjacent under . For a fixed positive integer , the graph contains all size- independent sets as its vertices/nodes, and two nodes are joined by an edge if their corresponding independent sets are adjacent under . A graph is called a -graph if there exists a graph such that and are isomorphic.
What is the complexity of Token Sliding on outerplanar graphs? And more generally, on series-parallel graphs ( graphs of treewidth at most two)?
Let be a graph. What are necessary and sufficient conditions for such that is acyclic?
Let be a forest. Then is a forest if and only if is -free, for some integer .
Independent Set Reconfiguration (ISR) is one of the most well-studied problems in the Combinatorial Reconfiguration [12, 5]. Among several variants of ISR, Token Sliding is of particular interest. One of the main reasons is that in Token Sliding one often needs to deal with the situation where a token must make “detours” by “moving away” (and then moving back later) to allow some other token to move . Alternatively, this can be seen as a kind of “bottleneck effect”  where a token may not be able to “reach” some vertices which are “far apart” from all tokens because some tokens “block the way”.
Our motivation for Open Problem 1 comes from the following two results that both appeared in 2014.
There exists a constant such that Token Sliding (and two other variants of ISR) is -complete on graphs of bandwidth at most .
Indeed, since a graph of bandwidth at most also has pathwidth and treewidth at most , Theorem 4 holds for graphs of pathwidth/treewidth at most . Moreover, for ,
Token Sliding is in on trees.
Naturally, on graphs of treewidth at most , one may ask what the value of that separates “hard” from “easy” is. Toward answering this question, a first step is probably to consider .
Our motivation for Open Problem 2 and Conjecture 3 comes from [1, 2]. In , the authors initiated the study of from a purely graph-theoretic viewpoint. They continued their study in  focusing on those that are acyclic. Open Problem 2 involves characterizing general graphs, which seems to be quite challenging and a first step may be to consider well-known graph classes. Toward this direction, in , the authors proved a forbidden induced subgraph characterization of a tree satisfying that is acyclic where and Conjecture 3, if true, will provide a complete analysis of which trees having acyclic -graphs.
For a fixed integer , a -path vertex cover of a graph is a vertex subset such that any path on vertices contains at least one member from . Two -path vertex covers are adjacent under Token Sliding () if there exists such that , , and . The -Path Vertex Cover Reconfiguration under Token Sliding (-PVCR-TS) problem asks if there is a -sequence between two given -path vertex covers and in , that is, a sequence of -path vertex covers starting with and ending with where any two consecutive members are adjacent under .
What is the complexity of -PVCR-TS on bipartite graphs? Or more restrictedly, on trees?
The -PVCR-TS problem was first introduced in . The complexities of -PVCR-TS and some other variants have been characterized in  for some graph classes, including planar and bounded bandwidth graphs, trees, paths, and cycles. However, -PVCR-TS on trees remains open. Our motivation comes from an attempt in  to solve this problem for a subclass of trees called caterpillars (i.e., trees obtained by attaching leaves to a central path). Unlike the cases for independent sets, a token may have to make “detours” by “moving closer” (and then moving back later) to allow some other tokens to move.
For a fixed integer , a proper -coloring of a graph is a function such that for any edge we have . If has a proper -coloring, we say that it is -colorable. Two proper -colorings are adjacent under Token Swapping if there exist such that , , for any , and . The -Recoloring under Token Swapping (-RTS) problem asks if there is a sequence of adjacent proper -colorings of under Token Swapping between two given proper -colorings . The graph takes all proper -colorings of as its nodes and their adjacency are defined under Token Swapping.
What is the complexity of -RTS on trees for some fixed ?
What is the smallest value of such that is connected for some given graph ?
What are necessary and sufficient conditions for a -colorable graph such that is connected?
The -RTS problem lies between two of the most well-studied problems in Combinatorial Reconfiguration: Token Swapping  and Vertex Recoloring . (See [12, 11] for more details.) RTS was first proposed in  where the authors claimed that RTS remains -complete on planar graphs and can be solved in polynomial time on paths () and cographs (any ). Up to present, their results have not yet been published. The goal of our proposed open problems is to obtain a better understanding on the similarities and differences between and the well-stuided graph —the graph whose nodes are proper -colorings of and two nodes are adjacent if one can be obtained from the other by recoloring exactly one vertex.