Discussiones Mathematicae Graph Theory最新文献_第6页

Efficient (j, k)-Dominating Functions 有效（j，k）-支配函数

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2355

W. Klostermeyer, G. MacGillivray, S. Semnani, Farzaneh Piri

{"title":"Efficient (j, k)-Dominating Functions","authors":"W. Klostermeyer, G. MacGillivray, S. Semnani, Farzaneh Piri","doi":"10.7151/dmgt.2355","DOIUrl":"https://doi.org/10.7151/dmgt.2355","url":null,"abstract":"Abstract For positive integers j and k, an efficient (j, k)-dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2, . . ., j} such that the sum of function values in the closed neighbourhood of every vertex equals k. The relationship between the existence of efficient (j, k)-dominating functions and various kinds of efficient dominating sets is explored. It is shown that if a strongly chordal graph has an efficient (j, k)-dominating function, then it has an efficient dominating set. Further, every efficient (j, k)-dominating function of a strongly chordal graph can be expressed as a sum of characteristic functions of efficient dominating sets. For j < k there are strongly chordal graphs with an efficient dominating set but no efficient (j, k)-dominating function. The problem of deciding whether a given graph has an efficient (j, k)-dominating function is shown to be NP-complete for all positive integers j and k, and solvable in polynomial time for strongly chordal graphs when j = k. By taking j = 1 we obtain NP-completeness of the problem of deciding whether a given graph has an efficient k-tuple dominating set for any fixed positive integer k. Finally, we consider efficient (2, 2)-dominating functions of trees. We describe a new constructive characterization of the trees with an efficient dominating set and a constructive characterization of the trees with two different efficient dominating sets. A number of open problems and questions are stated throughout the work.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49363110","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}

引用次数: 0

The Hilton-Spencer Cycle Theorems Via Katona’s Shadow Intersection Theorem Hilton—Spencer循环定理与Katona的阴影交定理

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2365

P. Borg, Carl Feghali

{"title":"The Hilton-Spencer Cycle Theorems Via Katona’s Shadow Intersection Theorem","authors":"P. Borg, Carl Feghali","doi":"10.7151/dmgt.2365","DOIUrl":"https://doi.org/10.7151/dmgt.2365","url":null,"abstract":"Abstract A family 𝒜 of sets is said to be intersecting if every two sets in 𝒜 intersect. An intersecting family is said to be trivial if its sets have a common element. A graph G is said to be r-EKR if at least one of the largest intersecting families of independent r-element sets of G is trivial. Let α (G) and ω (G) denote the independence number and the clique number of G, respectively. Hilton and Spencer recently showed that if G is the vertex-disjoint union of a cycle C raised to the power k and s cycles 1C, . . ., sC raised to the powers k1, . . ., ks, respectively, 1 ≤ r ≤ α (G), and min(ω(C1k1),…,ω(Csks))≥ω(Ck), min left( {omega left( {{}_1{C^{k1}}} right), ldots ,omega left( {{}_s{C^{ks}}} right)} right) ge omega left( {{C^k}} right), then G is r-EKR. They had shown that the same holds if C is replaced by a path P and the condition on the clique numbers is relaxed to min(ω(C1k1),…,ω(Csks))≥ω(Pk), min left( {omega left( {{}_1{C^{k1}}} right), ldots ,omega left( {{}_s{C^{ks}}} right)} right) ge omega left( {{P^k}} right), We use the classical Shadow Intersection Theorem of Katona to obtain a significantly shorter proof of each result for the case where the inequality for the minimum clique number is strict.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48899381","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}

引用次数: 0

The Existence of Path-Factor Covered Graphs 路径因子覆盖图的存在性

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2353

Guowei Dai

引用次数: 10

Burnside Chromatic Polynomials of Group-Invariant Graphs 群不变图的Burnside色多项式

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2385

J. White

引用次数: 0

Relaxed DP-Coloring and another Generalization of DP-Coloring on Planar Graphs without 4-Cycles and 7-Cycles 无4环和7环平面图上的松弛DP染色和DP染色的另一个推广

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2405

Sarawute Sribunhung, K. Nakprasit, Kittikorn Nakprasit, Pongpat Sittitrai

{"title":"Relaxed DP-Coloring and another Generalization of DP-Coloring on Planar Graphs without 4-Cycles and 7-Cycles","authors":"Sarawute Sribunhung, K. Nakprasit, Kittikorn Nakprasit, Pongpat Sittitrai","doi":"10.7151/dmgt.2405","DOIUrl":"https://doi.org/10.7151/dmgt.2405","url":null,"abstract":"Abstract DP-coloring is generalized via relaxed coloring and variable degeneracy in [P. Sittitrai and K. Nakprasit, Su cient conditions on planar graphs to have a relaxed DP-3-coloring, Graphs Combin. 35 (2019) 837–845], [K.M. Nakprasit and K. Nakprasit, A generalization of some results on list coloring and DP-coloring, Graphs Combin. 36 (2020) 1189–1201] and [P. Sittitrai and K. Nakprasit, An analogue of DP-coloring for variable degeneracy and its applications, Discuss. Math. Graph Theory]. In this work, we introduce another concept that includes two previous generalizations. We demonstrate its application on planar graphs without 4-cycles and 7-cycles. One implication is that the vertex set of every planar graph without 4-cycles and 7-cycles can be partitioned into three sets in which each of them induces a linear forest and one of them is an independent set. Additionally, we show that every planar graph without 4-cycles and 7-cycles is DP-(1, 1, 1)-colorable. This generalizes a result of Lih et al. [A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001) 269–273] that every planar graph without 4-cycles and 7-cycles is (3, 1)*-choosable.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45542082","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}

引用次数: 3

Forbidden Subgraphs for Existences of (Connected) 2-Factors of a Graph 图的(连通)2因子存在的禁止子图

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2366

Xiaojing Yang, Liming Xiong

引用次数: 0

Unique Minimum Semipaired Dominating Sets in Trees 树中的唯一最小半对支配集

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2349

T. Haynes, Michael A. Henning

引用次数: 1

Extending Potočnik and Šajna’s Conditions on the Existence of Vertex-Transitive Self-Complementary k-Hypergraphs 推广poto<e:1> nik和Šajna关于顶点传递自互补k-超图存在的条件

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2360

L. Lesniak, Henri Thuiller, A. Wojda

引用次数: 0

A Note on the Upper Bounds on the Size of Bipartite and Tripartite 1-Embeddable Graphs on Surfaces 曲面上二部和三部1-可嵌入图大小上界的一个注记

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2361

Hikari Shibuya, Yusuke Suzuki

引用次数: 0

Equimatchable Bipartite Graphs 等匹配二部图

IF 0.7 4区数学

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI: 10.7151/dmgt.2356

Yasemin Büyükçolak, Didem Gözüpek, Sibel Ozkan

{"title":"Equimatchable Bipartite Graphs","authors":"Yasemin Büyükçolak, Didem Gözüpek, Sibel Ozkan","doi":"10.7151/dmgt.2356","DOIUrl":"https://doi.org/10.7151/dmgt.2356","url":null,"abstract":"Abstract A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [Equi-matchable graphs, Graph Theory and Combinatorics (Academic Press, London, 1984) 239–254] has provided a characterization of equimatchable bipartite graphs. Motivated by the fact that this characterization is not structural, Frendrup et al. [A note on equimatchable graphs, Australas. J. Combin. 46 (2010) 185–190] has also provided a structural characterization for equimatchable graphs with girth at least five, in particular, a characterization for equimatchable bipartite graphs with girth at least six. In this paper, we extend the characterization of Frendrup by eliminating the girth condition. For an equimatchable graph, an edge is said to be a critical-edge if the graph obtained by the removal of this edge is not equimatchable. An equimatchable graph is called edge-critical, denoted by ECE, if every edge is critical. Noting that each ECE-graph can be obtained from some equimatchable graph by recursively removing non-critical edges, each equimatchable graph can also be constructed from some ECE-graph by joining some non-adjacent vertices. Our study reduces the characterization of equimatchable bipartite graphs to the characterization of bipartite ECE-graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44276827","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}

引用次数: 0