W. Klostermeyer, G. MacGillivray, S. Semnani, Farzaneh Piri
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.
{"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}
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.
{"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}
Abstract A spanning subgraph H of a graph G is called a P≥k-factor of G if every component of H is isomorphic to a path of order at least k, where k ≥ 2. A graph G is called a P≥k-factor covered graph if there is a P≥k-factor of G covering e for any e ∈ E(G). In this paper, we obtain two special classes of P≥2-factor covered graphs. We also obtain two special classes of P≥3-factor covered graphs. Furthermore, it is shown that these results are all sharp.
{"title":"The Existence of Path-Factor Covered Graphs","authors":"Guowei Dai","doi":"10.7151/dmgt.2353","DOIUrl":"https://doi.org/10.7151/dmgt.2353","url":null,"abstract":"Abstract A spanning subgraph H of a graph G is called a P≥k-factor of G if every component of H is isomorphic to a path of order at least k, where k ≥ 2. A graph G is called a P≥k-factor covered graph if there is a P≥k-factor of G covering e for any e ∈ E(G). In this paper, we obtain two special classes of P≥2-factor covered graphs. We also obtain two special classes of P≥3-factor covered graphs. Furthermore, it is shown that these results are all sharp.","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":"47339155","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 We introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group 𝕲 acting on a graph G and a 𝕲-set X, a proper X-coloring is a function with no monochromatic edge orbit. The set of proper colorings is a 𝕲-set which induces a polynomial function from the Burnside ring of 𝕲 to itself. In this paper, we study many properties of the Burnside chromatic polynomial, answering some questions of Zaslavsky.
{"title":"Burnside Chromatic Polynomials of Group-Invariant Graphs","authors":"J. White","doi":"10.7151/dmgt.2385","DOIUrl":"https://doi.org/10.7151/dmgt.2385","url":null,"abstract":"Abstract We introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group 𝕲 acting on a graph G and a 𝕲-set X, a proper X-coloring is a function with no monochromatic edge orbit. The set of proper colorings is a 𝕲-set which induces a polynomial function from the Burnside ring of 𝕲 to itself. In this paper, we study many properties of the Burnside chromatic polynomial, answering some questions of Zaslavsky.","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":"43891078","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}
Sarawute Sribunhung, K. Nakprasit, Kittikorn Nakprasit, Pongpat Sittitrai
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.
{"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}
Abstract Clearly, having a 2-factor in a graph is a necessary condition for a graph to be hamiltonian, while having an even factor in graph is a necessary condition for a graph to have a 2-factor. In this paper, we completely characterize the forbidden subgraph and pairs of forbidden subgraphs that force a 2-connected graph admitting a 2-factor (a necessary condition) to be hamiltonian and a connected graph with an even factor (a necessary condition) to have a 2-factor, respectively. Our results show that these pairs of forbidden subgraphs become wider than those in Faudree, Gould and in Fujisawa, Saito, respectively, if we impose the two necessary conditions, respectively.
{"title":"Forbidden Subgraphs for Existences of (Connected) 2-Factors of a Graph","authors":"Xiaojing Yang, Liming Xiong","doi":"10.7151/dmgt.2366","DOIUrl":"https://doi.org/10.7151/dmgt.2366","url":null,"abstract":"Abstract Clearly, having a 2-factor in a graph is a necessary condition for a graph to be hamiltonian, while having an even factor in graph is a necessary condition for a graph to have a 2-factor. In this paper, we completely characterize the forbidden subgraph and pairs of forbidden subgraphs that force a 2-connected graph admitting a 2-factor (a necessary condition) to be hamiltonian and a connected graph with an even factor (a necessary condition) to have a 2-factor, respectively. Our results show that these pairs of forbidden subgraphs become wider than those in Faudree, Gould and in Fujisawa, Saito, respectively, if we impose the two necessary conditions, respectively.","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":"47882113","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 Let G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.
{"title":"Unique Minimum Semipaired Dominating Sets in Trees","authors":"T. Haynes, Michael A. Henning","doi":"10.7151/dmgt.2349","DOIUrl":"https://doi.org/10.7151/dmgt.2349","url":null,"abstract":"Abstract Let G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.","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":"45837397","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 Let ℓ be a positive integer, k = 2ℓ or k = 2ℓ + 1, and let n be a positive integer with n ≡ 1 (mod 2ℓ+1). For a prime p, n(p) denotes the largest integer i such that pi divides n. Potočnik and Šajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have pn(p) ≡ 1 (mod 2ℓ+1). Here we extend their result to a larger class of integers k.
{"title":"Extending Potočnik and Šajna’s Conditions on the Existence of Vertex-Transitive Self-Complementary k-Hypergraphs","authors":"L. Lesniak, Henri Thuiller, A. Wojda","doi":"10.7151/dmgt.2360","DOIUrl":"https://doi.org/10.7151/dmgt.2360","url":null,"abstract":"Abstract Let ℓ be a positive integer, k = 2ℓ or k = 2ℓ + 1, and let n be a positive integer with n ≡ 1 (mod 2ℓ+1). For a prime p, n(p) denotes the largest integer i such that pi divides n. Potočnik and Šajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have pn(p) ≡ 1 (mod 2ℓ+1). Here we extend their result to a larger class of integers k.","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":"41530633","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 note, we show sharp upper bounds of the size of simple bipartite and tripartite 1-embeddable graphs on closed surfaces.
摘要在本文中，我们给出了闭曲面上简单二分图和三分1-可嵌入图的大小的尖锐上界。
{"title":"A Note on the Upper Bounds on the Size of Bipartite and Tripartite 1-Embeddable Graphs on Surfaces","authors":"Hikari Shibuya, Yusuke Suzuki","doi":"10.7151/dmgt.2361","DOIUrl":"https://doi.org/10.7151/dmgt.2361","url":null,"abstract":"Abstract In this note, we show sharp upper bounds of the size of simple bipartite and tripartite 1-embeddable graphs on closed surfaces.","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":"43496641","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 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.
{"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}
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