Abstract A digraph D is supereulerian if D contains a spanning eulerian subdigraph. In this paper, we propose the following problem: is there an integer t with 0 ≤ t ≤ n − 3 so that any strong digraph with n vertices satisfying either both d(u) ≥ n − 1 + t and d(v) ≥ n − 2 − t or both d(u) ≥ n − 2 − t and d(v) ≥ n − 1 + t, for any pair of dominated or dominating nonadjacent vertices {u, v}, is supereulerian? We prove the cases when t = 0, t = n − 4 and t = n − 3. Moreover, we show that if a strong digraph D with n vertices satisfies min{d+(u)+d−(v), d−(u)+d+(v)} ≥ n−1 for any pair of dominated or dominating nonadjacent vertices {u, v} of D, then D is supereulerian.
如果有向图D包含一个生成欧拉子图，则D是超欧拉图。在本文中，我们提出了以下问题:是否存在一个0≤t≤n−3的整数t，使得任何有n个顶点的强有向图既满足d(u)≥n−1 + t又满足d(v)≥n−2 - t或者d(u)≥n−2 - t又满足d(v)≥n−1 + t，对于任意支配或支配的非相邻顶点{u, v}，都是超欧拉图?我们证明了t = 0, t = n - 4和t = n - 3的情况。此外，我们证明了如果一个有n个顶点的强有向图D满足min{D +(u)+ D−(v)， D−(u)+ D +(v)}≥n−1，对于D的任意对支配或支配的非相邻顶点{u, v}，则D是超欧拉的。
{"title":"Dominated Pair Degree Sum Conditions of Supereulerian Digraphs","authors":"Changchang Dong, J. Meng, Juan Liu","doi":"10.7151/dmgt.2476","DOIUrl":"https://doi.org/10.7151/dmgt.2476","url":null,"abstract":"Abstract A digraph D is supereulerian if D contains a spanning eulerian subdigraph. In this paper, we propose the following problem: is there an integer t with 0 ≤ t ≤ n − 3 so that any strong digraph with n vertices satisfying either both d(u) ≥ n − 1 + t and d(v) ≥ n − 2 − t or both d(u) ≥ n − 2 − t and d(v) ≥ n − 1 + t, for any pair of dominated or dominating nonadjacent vertices {u, v}, is supereulerian? We prove the cases when t = 0, t = n − 4 and t = n − 3. Moreover, we show that if a strong digraph D with n vertices satisfies min{d+(u)+d−(v), d−(u)+d+(v)} ≥ n−1 for any pair of dominated or dominating nonadjacent vertices {u, v} of D, then D is supereulerian.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48254767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a graph G, a k-labelling ℓ of G is an assignment ℓ : E(G) → {1, . . . , k} of labels from {1, . . . , k} to the edges. We say that ℓ is s-proper, m-proper or p-proper, if no two adjacent vertices of G are incident to the same sum, multiset or product, respectively, of labels. Proper labellings are part of the field of distinguishing labellings, and have been receiving quite some attention over the last decades, in particular in the context of the well-known 1-2-3 Conjecture. In recent years, quite some progress was made towards the main questions of the field, with, notably, the analogues of the 1-2-3 Conjecture for m-proper and p-proper labellings being solved. This followed mainly from a better global understanding of these types of labellings. In this note, we focus on a question raised by Paramaguru and Sampathkumar, who asked whether graphs with m-proper 2-labellings always admit s-proper 2-labellings. A negative answer to this question was recently given by Luiz, who provided infinite families of counterexamples. We give a more general result, showing that recognising graphs with m-proper 2-labellings but no s-proper 2-labellings is an NP-hard problem. We also prove a similar result for m-proper 2-labellings and p-proper 2-labellings, and raise a few directions for further work on the topic.
{"title":"On Proper 2-Labellings Distinguishing by Sums, Multisets or Products","authors":"Julien Bensmail, Foivos Fioravantes","doi":"10.7151/dmgt.2473","DOIUrl":"https://doi.org/10.7151/dmgt.2473","url":null,"abstract":"Abstract Given a graph G, a k-labelling ℓ of G is an assignment ℓ : E(G) → {1, . . . , k} of labels from {1, . . . , k} to the edges. We say that ℓ is s-proper, m-proper or p-proper, if no two adjacent vertices of G are incident to the same sum, multiset or product, respectively, of labels. Proper labellings are part of the field of distinguishing labellings, and have been receiving quite some attention over the last decades, in particular in the context of the well-known 1-2-3 Conjecture. In recent years, quite some progress was made towards the main questions of the field, with, notably, the analogues of the 1-2-3 Conjecture for m-proper and p-proper labellings being solved. This followed mainly from a better global understanding of these types of labellings. In this note, we focus on a question raised by Paramaguru and Sampathkumar, who asked whether graphs with m-proper 2-labellings always admit s-proper 2-labellings. A negative answer to this question was recently given by Luiz, who provided infinite families of counterexamples. We give a more general result, showing that recognising graphs with m-proper 2-labellings but no s-proper 2-labellings is an NP-hard problem. We also prove a similar result for m-proper 2-labellings and p-proper 2-labellings, and raise a few directions for further work on the topic.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43164988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we study domination between different types of walks connecting two non-adjacent vertices of a graph. In particular, we center our attention on weakly toll walk and lk-path for k ∈ {2, 3}. A walk between two non-adjacent vertices in a graph G is called a weakly toll walk if the first and the last vertices in the walk are adjacent, respectively, only to the second and second-to-last vertices, which may occur more than once in the walk. And an lk-path is an induced path of length at most k between two non-adjacent vertices in a graph G. We study the domination between weakly toll walks, lk-paths (k ∈ {2, 3}) and different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks, weakly toll walks, lk-paths for k ∈ {3, 4}), and show how these give rise to characterizations of graph classes.
{"title":"On Walk Domination: Weakly Toll Domination, l2 and l3 Domination","authors":"M. Gutierrez, S. Tondato","doi":"10.7151/dmgt.2475","DOIUrl":"https://doi.org/10.7151/dmgt.2475","url":null,"abstract":"Abstract In this paper we study domination between different types of walks connecting two non-adjacent vertices of a graph. In particular, we center our attention on weakly toll walk and lk-path for k ∈ {2, 3}. A walk between two non-adjacent vertices in a graph G is called a weakly toll walk if the first and the last vertices in the walk are adjacent, respectively, only to the second and second-to-last vertices, which may occur more than once in the walk. And an lk-path is an induced path of length at most k between two non-adjacent vertices in a graph G. We study the domination between weakly toll walks, lk-paths (k ∈ {2, 3}) and different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks, weakly toll walks, lk-paths for k ∈ {3, 4}), and show how these give rise to characterizations of graph classes.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47467524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The generalized connectivity, an extension of connectivity, provides a new reference for measuring the fault tolerance of networks. For any connected graph G, let S ⊆ V (G) and 2 ≤ |S| ≤ V (G); κG(S) refers to the maximum number of internally disjoint trees in G connecting S. The generalized k-connectivity of G, κk(G), is defined as the minimum value of κG(S) over all S ⊆ V (G) with |S| = k. The n-dimensional crossed cube CQn, as a hypercube-like network, is considered as an attractive alternative to hypercube network because of its many good properties. In this paper, we study the generalized 3-connectivity and the generalized 4-connectivity of CQnand obtain κ3(CQn) = κ4(CQn) = n − 1, where n ≥ 2.
广义连通性是连通性的一种扩展，为网络容错能力的测量提供了新的参考。对任意连通图G，令S≤|S|≤V (G);κG(S)是连接S的G中内部不相交树的最大个数。G的广义k-连通性，κk(G)被定义为κG(S)在所有S的 V (G)上的最小值，且|S| = k。n维交叉立方体CQn作为一种超立方体网络，由于其许多良好的性质，被认为是超立方体网络的一种有吸引力的替代方案。研究了CQn的广义3-连通性和广义4-连通性，得到了κ3(CQn) = κ4(CQn) = n−1，其中n≥2。
{"title":"The Generalized 3-Connectivity and 4-Connectivity of Crossed Cube","authors":"Heqin Liu, Dongqin Cheng","doi":"10.7151/dmgt.2474","DOIUrl":"https://doi.org/10.7151/dmgt.2474","url":null,"abstract":"Abstract The generalized connectivity, an extension of connectivity, provides a new reference for measuring the fault tolerance of networks. For any connected graph G, let S ⊆ V (G) and 2 ≤ |S| ≤ V (G); κG(S) refers to the maximum number of internally disjoint trees in G connecting S. The generalized k-connectivity of G, κk(G), is defined as the minimum value of κG(S) over all S ⊆ V (G) with |S| = k. The n-dimensional crossed cube CQn, as a hypercube-like network, is considered as an attractive alternative to hypercube network because of its many good properties. In this paper, we study the generalized 3-connectivity and the generalized 4-connectivity of CQnand obtain κ3(CQn) = κ4(CQn) = n − 1, where n ≥ 2.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46274625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. A graph is said to be k(≥1)-extendable if every matching of size k can be extended to a perfect matching. It is known that the vertex connectivity of a 1-plane graph is at most 7. In this paper, we characterize the k-extendability of 7-connected maximal 1-plane graphs. We show that every 7-connected maximal 1-plane graph with even order is k-extendable for 1 ≤ k ≤ 3. And any 7-connected maximal 1-plane graph is not k-extendable for 4 ≤ k ≤ 11. As for k ≥ 12, any 7-connected maximal 1-plane graph with n vertices is not k-extendable unless n = 2k.
{"title":"The Matching Extendability of 7-Connected Maximal 1-Plane Graphs","authors":"Yuanqiu Huang, Licheng Zhang, Yuxi Wang","doi":"10.7151/dmgt.2470","DOIUrl":"https://doi.org/10.7151/dmgt.2470","url":null,"abstract":"Abstract A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. A graph is said to be k(≥1)-extendable if every matching of size k can be extended to a perfect matching. It is known that the vertex connectivity of a 1-plane graph is at most 7. In this paper, we characterize the k-extendability of 7-connected maximal 1-plane graphs. We show that every 7-connected maximal 1-plane graph with even order is k-extendable for 1 ≤ k ≤ 3. And any 7-connected maximal 1-plane graph is not k-extendable for 4 ≤ k ≤ 11. As for k ≥ 12, any 7-connected maximal 1-plane graph with n vertices is not k-extendable unless n = 2k.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45622765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A graph G of order n is arbitrarily partitionable (AP for short) if, for every partition (λ1, . . ., λp) of n, there is a partition (V1, . . ., Vp) of V (G) such that G[Vi] is a connected graph of order λi for every i ∈ {1, . . ., p}. Several aspects of AP graphs have been investigated to date, including their connection to Hamiltonian graphs and traceable graphs. Every traceable graph (and, thus, Hamiltonian graph) is indeed known to be AP, and a line of research on AP graphs is thus about weakening, to APness, known sufficient conditions for graphs to be Hamiltonian or traceable. In this work, we provide a sufficient condition for APness involving the parameter ̄σ3, where, for a given graph G, the parameter ̄σ3(G) is defined as the minimum value of d(u) + d(v) + d(w) − |N(u) ∩ N(v) ∩ N(w)| for a set {u, v, w} of three pairwise independent vertices u, v, and w of G. Flandrin, Jung, and Li proved that any graph G of order n is Hamitonian provided G is 2-connected and ̄σ3(G) ≥ n, and traceable provided ̄σ3(G) ≥ n − 1. Unfortunately, we exhibit examples showing that having ̄σ3(G) ≥ n − 2 is not a guarantee for G to be AP. However, we prove that G is AP provided G is 2-connected, ̄σ3(G) ≥ n−2, and G has a perfect matching or quasi-perfect matching.
{"title":"A σ3 Condition for Arbitrarily Partitionable Graphs","authors":"Julien Bensmail","doi":"10.7151/dmgt.2471","DOIUrl":"https://doi.org/10.7151/dmgt.2471","url":null,"abstract":"Abstract A graph G of order n is arbitrarily partitionable (AP for short) if, for every partition (λ1, . . ., λp) of n, there is a partition (V1, . . ., Vp) of V (G) such that G[Vi] is a connected graph of order λi for every i ∈ {1, . . ., p}. Several aspects of AP graphs have been investigated to date, including their connection to Hamiltonian graphs and traceable graphs. Every traceable graph (and, thus, Hamiltonian graph) is indeed known to be AP, and a line of research on AP graphs is thus about weakening, to APness, known sufficient conditions for graphs to be Hamiltonian or traceable. In this work, we provide a sufficient condition for APness involving the parameter ̄σ3, where, for a given graph G, the parameter ̄σ3(G) is defined as the minimum value of d(u) + d(v) + d(w) − |N(u) ∩ N(v) ∩ N(w)| for a set {u, v, w} of three pairwise independent vertices u, v, and w of G. Flandrin, Jung, and Li proved that any graph G of order n is Hamitonian provided G is 2-connected and ̄σ3(G) ≥ n, and traceable provided ̄σ3(G) ≥ n − 1. Unfortunately, we exhibit examples showing that having ̄σ3(G) ≥ n − 2 is not a guarantee for G to be AP. However, we prove that G is AP provided G is 2-connected, ̄σ3(G) ≥ n−2, and G has a perfect matching or quasi-perfect matching.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42234706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. de Wet, M. Frick, O. Oellermann, Jean E. Dunbar
Abstract The detour order of a graph G, denoted by τ (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that τ (〈A〉) ≤ a and τ (〈B〉) ≤ b, we say that (A, B) is an (a, b)-partition of G. If equality holds in both instances, we call (A, B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that τ (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a ≤ 8.
{"title":"On the Strong Path Partition Conjecture","authors":"J. de Wet, M. Frick, O. Oellermann, Jean E. Dunbar","doi":"10.7151/dmgt.2468","DOIUrl":"https://doi.org/10.7151/dmgt.2468","url":null,"abstract":"Abstract The detour order of a graph G, denoted by τ (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that τ (〈A〉) ≤ a and τ (〈B〉) ≤ b, we say that (A, B) is an (a, b)-partition of G. If equality holds in both instances, we call (A, B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that τ (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a ≤ 8.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47047431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A well-known theorem of Vizing separates graphs into two classes: those which admit proper Δ-edge-colourings, known as class one graphs; and those which do not, known as class two graphs. Class two graphs do admit proper (Δ+ 1)-edge-colourings. In the context of snarks (class two cubic graphs), there has recently been much focus on parameters which are said to measure how far the snark is from being 3-edge-colourable, and there are thus many well-known lemmas and results which are widely used in the study of snarks. These parameters, or so-called measurements of uncolourability, have thus far evaded consideration in the general case of k-regular class two graphs for k > 3. Two such measures are the resistance and vertex resistance of a graph. For a graph G, the (vertex) resistance of G, denoted as (rv(G)) r(G), is defined as the minimum number of (vertices) edges which need to be removed from G in order to render it class one. In this paper, we generalise some of the well-known lemmas and results to the k-regular case. For the main result of this paper, we generalise the known fact that r(G) = rv(G) if G is a snark by proving the following bounds for k-regular G:rv(G)≤r(G)≤⌊ k2 ⌋rv(G) G:{r_v}left( G right) le rleft( G right) le leftlfloor {{k over 2}} rightrfloor {r_v}left( G right) . Moreover, we show that both bounds are best possible for any even k.
{"title":"Resistance in Regular Class Two Graphs","authors":"I. Allie, Jordan Arenstein","doi":"10.7151/dmgt.2467","DOIUrl":"https://doi.org/10.7151/dmgt.2467","url":null,"abstract":"Abstract A well-known theorem of Vizing separates graphs into two classes: those which admit proper Δ-edge-colourings, known as class one graphs; and those which do not, known as class two graphs. Class two graphs do admit proper (Δ+ 1)-edge-colourings. In the context of snarks (class two cubic graphs), there has recently been much focus on parameters which are said to measure how far the snark is from being 3-edge-colourable, and there are thus many well-known lemmas and results which are widely used in the study of snarks. These parameters, or so-called measurements of uncolourability, have thus far evaded consideration in the general case of k-regular class two graphs for k > 3. Two such measures are the resistance and vertex resistance of a graph. For a graph G, the (vertex) resistance of G, denoted as (rv(G)) r(G), is defined as the minimum number of (vertices) edges which need to be removed from G in order to render it class one. In this paper, we generalise some of the well-known lemmas and results to the k-regular case. For the main result of this paper, we generalise the known fact that r(G) = rv(G) if G is a snark by proving the following bounds for k-regular G:rv(G)≤r(G)≤⌊ k2 ⌋rv(G) G:{r_v}left( G right) le rleft( G right) le leftlfloor {{k over 2}} rightrfloor {r_v}left( G right) . Moreover, we show that both bounds are best possible for any even k.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44092524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If $X=(V(X),E(X))$ and $Y=(V(Y),E(Y))$ are $n$-vertex graphs, then their friends-and-strangers graph $mathsf{FS}(X,Y)$ is the graph whose vertices are the bijections from $V(X)$ to $V(Y)$ in which two bijections $sigma$ and $sigma'$ are adjacent if and only if there is an edge ${a,b}in E(X)$ such that ${sigma(a),sigma(b)}in E(Y)$ and $sigma'=sigmacirc (a,,b)$, where $(a,,b)$ is the permutation of $V(X)$ that swaps $a$ and $b$. We prove general theorems that provide necessary and/or sufficient conditions for $mathsf{FS}(X,Y)$ to be connected. As a corollary, we obtain a complete characterization of the graphs $Y$ such that $mathsf{FS}(mathsf{Dand}_{k,n},Y)$ is connected, where $mathsf{Dand}_{k,n}$ is a dandelion graph; this substantially generalizes a theorem of the first author and Kravitz in the case $k=3$. For specific choices of $Y$, we characterize the spider graphs $X$ such that $mathsf{FS}(X,Y)$ is connected. In a different vein, we study the cycle spaces of friends-and-strangers graphs. Naatz proved that if $X$ is a path graph, then the cycle space of $mathsf{FS}(X,Y)$ is spanned by $4$-cycles and $6$-cycles; we show that the same statement holds when $X$ is a cycle and $Y$ has domination number at least $3$. When $X$ is a cycle and $Y$ has domination number at least $2$, our proof sheds light on how walks in $mathsf{FS}(X,Y)$ behave under certain Coxeter moves.
{"title":"Connectedness and cycle spaces of friends-and-strangers graphs","authors":"Colin Defant, David Dong, Alan Lee, Michelle Wei","doi":"10.7151/dmgt.2492","DOIUrl":"https://doi.org/10.7151/dmgt.2492","url":null,"abstract":"If $X=(V(X),E(X))$ and $Y=(V(Y),E(Y))$ are $n$-vertex graphs, then their friends-and-strangers graph $mathsf{FS}(X,Y)$ is the graph whose vertices are the bijections from $V(X)$ to $V(Y)$ in which two bijections $sigma$ and $sigma'$ are adjacent if and only if there is an edge ${a,b}in E(X)$ such that ${sigma(a),sigma(b)}in E(Y)$ and $sigma'=sigmacirc (a,,b)$, where $(a,,b)$ is the permutation of $V(X)$ that swaps $a$ and $b$. We prove general theorems that provide necessary and/or sufficient conditions for $mathsf{FS}(X,Y)$ to be connected. As a corollary, we obtain a complete characterization of the graphs $Y$ such that $mathsf{FS}(mathsf{Dand}_{k,n},Y)$ is connected, where $mathsf{Dand}_{k,n}$ is a dandelion graph; this substantially generalizes a theorem of the first author and Kravitz in the case $k=3$. For specific choices of $Y$, we characterize the spider graphs $X$ such that $mathsf{FS}(X,Y)$ is connected. In a different vein, we study the cycle spaces of friends-and-strangers graphs. Naatz proved that if $X$ is a path graph, then the cycle space of $mathsf{FS}(X,Y)$ is spanned by $4$-cycles and $6$-cycles; we show that the same statement holds when $X$ is a cycle and $Y$ has domination number at least $3$. When $X$ is a cycle and $Y$ has domination number at least $2$, our proof sheds light on how walks in $mathsf{FS}(X,Y)$ behave under certain Coxeter moves.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41527870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maya G. S. Thankachy, Ullas Chandran S.V., J. Tuite, Elias John Thomas, Gabriele Di Stefano, G. Erskine
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S subseteq V(G)$ is an emph{$x$-position set} if for any $y in S$ the shortest $x,y$-paths in $G$ contain no point of $Ssetminus { y}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.
{"title":"On the vertex position number of graphs","authors":"Maya G. S. Thankachy, Ullas Chandran S.V., J. Tuite, Elias John Thomas, Gabriele Di Stefano, G. Erskine","doi":"10.7151/dmgt.2491","DOIUrl":"https://doi.org/10.7151/dmgt.2491","url":null,"abstract":"In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S subseteq V(G)$ is an emph{$x$-position set} if for any $y in S$ the shortest $x,y$-paths in $G$ contain no point of $Ssetminus { y}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45534201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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