In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.
{"title":"High-ordered spectral characterization of unicyclic graphs","authors":"Yi-Zheng Fan, Hong Yang, Jian Zheng","doi":"10.7151/dmgt.2489","DOIUrl":"https://doi.org/10.7151/dmgt.2489","url":null,"abstract":"In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46551000","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 consider the problem of extending and avoiding partial edge colorings of trees; that is, given a partial edge coloring φ of a tree T we are interested in whether there is a proper Δ(T )-edge coloring of T that agrees with the coloring φ on every edge that is colored under φ; or, similarly, if there is a proper Δ(T )-edge coloring that disagrees with φ on every edge that is colored under φ. We characterize which partial edge colorings with at most Δ(T ) + 1 precolored edges in a tree T are extendable, thereby proving an analogue of a result by Andersen for Latin squares. Furthermore we obtain some “mixed” results on extending a partial edge coloring subject to the condition that the extension should avoid a given partial edge coloring; in particular, for all 0 ≤ k ≤ Δ(T ), we characterize for which configurations consisting of a partial coloring φ of Δ(T ) − k edges and a partial coloring ψ of k + 1 edges of a tree T, there is an extension of φ that avoids ψ.
{"title":"Edge Precoloring Extension of Trees II","authors":"C. J. Casselgren, F. B. Petros","doi":"10.7151/dmgt.2461","DOIUrl":"https://doi.org/10.7151/dmgt.2461","url":null,"abstract":"Abstract We consider the problem of extending and avoiding partial edge colorings of trees; that is, given a partial edge coloring φ of a tree T we are interested in whether there is a proper Δ(T )-edge coloring of T that agrees with the coloring φ on every edge that is colored under φ; or, similarly, if there is a proper Δ(T )-edge coloring that disagrees with φ on every edge that is colored under φ. We characterize which partial edge colorings with at most Δ(T ) + 1 precolored edges in a tree T are extendable, thereby proving an analogue of a result by Andersen for Latin squares. Furthermore we obtain some “mixed” results on extending a partial edge coloring subject to the condition that the extension should avoid a given partial edge coloring; in particular, for all 0 ≤ k ≤ Δ(T ), we characterize for which configurations consisting of a partial coloring φ of Δ(T ) − k edges and a partial coloring ψ of k + 1 edges of a tree T, there is an extension of φ that avoids ψ.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46389041","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 An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f, or (iii) vw = e or f. An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1, . . ., p} such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors. In this paper, we introduce and study the strong version of incidence colouring, where incidences adjacent to the same incidence must also get distinct colours. We determine the exact value of — or upper bounds on — the strong incidence chromatic number of several classes of graphs, namely cycles, wheel graphs, trees, ladder graphs, square grids and subclasses of Halin graphs.
{"title":"Strong Incidence Colouring of Graphs","authors":"Brahim Benmedjdoub, É. Sopena","doi":"10.7151/dmgt.2466","DOIUrl":"https://doi.org/10.7151/dmgt.2466","url":null,"abstract":"Abstract An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f, or (iii) vw = e or f. An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1, . . ., p} such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors. In this paper, we introduce and study the strong version of incidence colouring, where incidences adjacent to the same incidence must also get distinct colours. We determine the exact value of — or upper bounds on — the strong incidence chromatic number of several classes of graphs, namely cycles, wheel graphs, trees, ladder graphs, square grids and subclasses of Halin graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41925627","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 adopt the recently introduced concept of the bipartite-hole-number due to McDiarmid and Yolov, and extend their result on Hamiltonicity to other Hamiltonian properties of graphs with a large minimum degree in terms of this concept. An (s, t)-bipartite-hole in a graph G consists of two disjoint sets of vertices S and T with |S| = s and |T| = t such that E(S, T ) =∅. The bipartite-hole-number α˜(G) tilde alpha left( G right) is the maximum integer r such that G contains an (s, t)-bipartite-hole for every pair of nonnegative integers s and t with s + t = r. Our main results are that a graph G is traceable if δ(G)≥α˜(G)−1 delta left( G right) ge tilde alpha left( G right) - 1 , and Hamilton-connected if δ(G)≥α˜(G)+1 delta left( G right) ge tilde alpha left( G right) + 1 , both improving the analogues of Dirac’s Theorem for traceable and Hamilton-connected graphs.
{"title":"A Note on Minimum Degree, Bipartite Holes, and Hamiltonian Properties","authors":"Qiannan Zhou, H. Broersma, Ligong Wang, Yong Lu","doi":"10.7151/dmgt.2464","DOIUrl":"https://doi.org/10.7151/dmgt.2464","url":null,"abstract":"Abstract We adopt the recently introduced concept of the bipartite-hole-number due to McDiarmid and Yolov, and extend their result on Hamiltonicity to other Hamiltonian properties of graphs with a large minimum degree in terms of this concept. An (s, t)-bipartite-hole in a graph G consists of two disjoint sets of vertices S and T with |S| = s and |T| = t such that E(S, T ) =∅. The bipartite-hole-number α˜(G) tilde alpha left( G right) is the maximum integer r such that G contains an (s, t)-bipartite-hole for every pair of nonnegative integers s and t with s + t = r. Our main results are that a graph G is traceable if δ(G)≥α˜(G)−1 delta left( G right) ge tilde alpha left( G right) - 1 , and Hamilton-connected if δ(G)≥α˜(G)+1 delta left( G right) ge tilde alpha left( G right) + 1 , both improving the analogues of Dirac’s Theorem for traceable and Hamilton-connected graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49619218","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 set S of vertices in a graph G is an outer connected dominating set of G if every vertex in V S is adjacent to a vertex in S and the subgraph induced by V S is connected. The outer connected domination number of G, denoted by γ˜c(G) {tilde gamma _c}left( G right) , is the minimum cardinality of an outer connected dominating set of G. Zhuang [Domination and outer connected domination in maximal outerplanar graphs, Graphs Combin. 37 (2021) 2679–2696] recently proved that γ˜c(G)≤⌊ n+k4 ⌋ {tilde gamma _c}left( G right) le leftlfloor {{{n + k} over 4}} rightrfloor for any maximal outerplanar graph G of order n ≥ 3 with k vertices of degree 2 and posed a conjecture which states that G is a striped maximal outerplanar graph with γ˜c(G)≤⌊ n+24 ⌋ {tilde gamma _c}left( G right) le leftlfloor {{{n + 2} over 4}} rightrfloor if and only if G ∈ 𝒜, where 𝒜 consists of six special families of striped outerplanar graphs. We disprove the conjecture. Moreover, we show that the conjecture become valid under some additional property to the striped maximal outerplanar graphs. In addition, we extend the above theorem of Zhuang to all maximal K2,3-minor free graphs without K4 and all K4-minor free graphs.
如果V S中的每个顶点与S中的一个顶点相邻，且由V S引生的子图是连通的，则图G中的顶点集S是G的外连通支配集。G的外连通支配数，用γ ～ c(G)表示 {tilde gamma ＿c}left(g) right)，是G.庄的一个外连通支配集的最小基数[最大外平面图中的支配和外连通支配，图组合，37(2021)2679-2696]最近证明了γ ～ c(G)≤⌊n+k4⌋ {tilde gamma ＿c}left(g) right） le leftlfloor {{{N + k} over 4}} rightrfloor 对于任意n≥3阶、k个顶点为2度的极大外平面图G，提出了一个猜想，说明G是一个γ ～ c(G)≤⌊n+24⌋的条纹极大外平面图 {tilde gamma ＿c}left(g) right） le leftlfloor {{{N + 2} over 4}} rightrfloor 当且仅当G∈φ，其中φ由6个特殊的条纹外平面图族组成。我们推翻了这个猜想。此外，我们还证明了在条纹极大外平面图的一些附加性质下，这个猜想是成立的。此外，我们将庄的上述定理推广到所有极大K2图、没有K4的3次自由图和所有K4次自由图。
{"title":"Outer Connected Domination in Maximal Outerplanar Graphs and Beyond","authors":"Wei Yang, Baoyindureng Wu","doi":"10.7151/dmgt.2462","DOIUrl":"https://doi.org/10.7151/dmgt.2462","url":null,"abstract":"Abstract A set S of vertices in a graph G is an outer connected dominating set of G if every vertex in V S is adjacent to a vertex in S and the subgraph induced by V S is connected. The outer connected domination number of G, denoted by γ˜c(G) {tilde gamma _c}left( G right) , is the minimum cardinality of an outer connected dominating set of G. Zhuang [Domination and outer connected domination in maximal outerplanar graphs, Graphs Combin. 37 (2021) 2679–2696] recently proved that γ˜c(G)≤⌊ n+k4 ⌋ {tilde gamma _c}left( G right) le leftlfloor {{{n + k} over 4}} rightrfloor for any maximal outerplanar graph G of order n ≥ 3 with k vertices of degree 2 and posed a conjecture which states that G is a striped maximal outerplanar graph with γ˜c(G)≤⌊ n+24 ⌋ {tilde gamma _c}left( G right) le leftlfloor {{{n + 2} over 4}} rightrfloor if and only if G ∈ 𝒜, where 𝒜 consists of six special families of striped outerplanar graphs. We disprove the conjecture. Moreover, we show that the conjecture become valid under some additional property to the striped maximal outerplanar graphs. In addition, we extend the above theorem of Zhuang to all maximal K2,3-minor free graphs without K4 and all K4-minor free graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45622443","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}
Guoliang Hao, Shouliu Wei, S. M. Sheikholeslami, Xiaodan Chen
Abstract Let G be a simple graph of order n and let γgdR(G) be the global double Roman domination number of G. In this paper, we give some upper bounds on the global double Roman domination number of G. In particular, we completely characterize the graph G with γgdR(G) = 2n − 2 and γgdR(G) = 2n − 3. Our results answer a question posed by Shao et al. (2019).
{"title":"Bounds on the Global Double Roman Domination Number in Graphs","authors":"Guoliang Hao, Shouliu Wei, S. M. Sheikholeslami, Xiaodan Chen","doi":"10.7151/dmgt.2460","DOIUrl":"https://doi.org/10.7151/dmgt.2460","url":null,"abstract":"Abstract Let G be a simple graph of order n and let γgdR(G) be the global double Roman domination number of G. In this paper, we give some upper bounds on the global double Roman domination number of G. In particular, we completely characterize the graph G with γgdR(G) = 2n − 2 and γgdR(G) = 2n − 3. Our results answer a question posed by Shao et al. (2019).","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44254559","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}
{"title":"On weakly Turán-good graphs","authors":"Dániel Gerbner","doi":"10.7151/dmgt.2510","DOIUrl":"https://doi.org/10.7151/dmgt.2510","url":null,"abstract":"Given graphs $H$ and $F$ with $chi(H)<chi(F)$, we say that $H$ is weakly $F$-Tur'an-good if among $n$-vertex $F$-free graphs, a $(chi(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a complete bipartite subgraph $K$ such that each vertex of $H$ is adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Tur'an-good, improving a very recent asymptotic bound due to Grzesik, GyH ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Tur'an-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Tur'an-good. We also show examples of graphs that are $C_{2k+1}$-Tur'an-good but not $C_{2ell+1}$-Tur'an-good for every $k>ell$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48890941","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 determine the order of magnitude of the minimum clique cover of the edges of a binomial, r-uniform, random hypergraph G(r)(n, p), p fixed. In doing so, we combine the ideas from the proofs of the graph case (r = 2) in Frieze and Reed [Covering the edges of a random graph by cliques, Combinatorica 15 (1995) 489–497] and Guo, Patten, Warnke [Prague dimension of random graphs, manuscript submitted for publication].
{"title":"Covering the Edges of a Random Hypergraph by Cliques","authors":"V. Rödl, A. Rucinski","doi":"10.7151/dmgt.2431","DOIUrl":"https://doi.org/10.7151/dmgt.2431","url":null,"abstract":"Abstract We determine the order of magnitude of the minimum clique cover of the edges of a binomial, r-uniform, random hypergraph G(r)(n, p), p fixed. In doing so, we combine the ideas from the proofs of the graph case (r = 2) in Frieze and Reed [Covering the edges of a random graph by cliques, Combinatorica 15 (1995) 489–497] and Guo, Patten, Warnke [Prague dimension of random graphs, manuscript submitted for publication].","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49227128","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}
Hadi Alizadeh, Didem Gözüpek, Gülnaz Boruzanli Ekinci
Abstract The domination gap of a graph G is defined as the di erence between the maximum and minimum cardinalities of a minimal dominating set in G. The term well-dominated graphs referring to the graphs with domination gap zero, was first introduced by Finbow et al. [Well-dominated graphs: A collection of well-covered ones, Ars Combin. 25 (1988) 5–10]. In this paper, we focus on the graphs with domination gap one which we term almost well-dominated graphs. While the results by Finbow et al. have implications for almost well-dominated graphs with girth at least 8, we extend these results to (C3, C4, C5, C7)-free almost well-dominated graphs by giving a complete structural characterization for such graphs.
{"title":"(C3, C4, C5, C7)-Free Almost Well-Dominated Graphs","authors":"Hadi Alizadeh, Didem Gözüpek, Gülnaz Boruzanli Ekinci","doi":"10.7151/dmgt.2331","DOIUrl":"https://doi.org/10.7151/dmgt.2331","url":null,"abstract":"Abstract The domination gap of a graph G is defined as the di erence between the maximum and minimum cardinalities of a minimal dominating set in G. The term well-dominated graphs referring to the graphs with domination gap zero, was first introduced by Finbow et al. [Well-dominated graphs: A collection of well-covered ones, Ars Combin. 25 (1988) 5–10]. In this paper, we focus on the graphs with domination gap one which we term almost well-dominated graphs. While the results by Finbow et al. have implications for almost well-dominated graphs with girth at least 8, we extend these results to (C3, C4, C5, C7)-free almost well-dominated graphs by giving a complete structural characterization for such graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47958867","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 10-regular graph which does not contain any 4-cycles. In this paper, we prove that G can be decomposed into paths of length 5, such that every vertex is a terminal of exactly two paths.
{"title":"Decomposing 10-Regular Graphs into Paths of Length 5","authors":"Mengmeng Xie, Chuixiang Zhou","doi":"10.7151/dmgt.2334","DOIUrl":"https://doi.org/10.7151/dmgt.2334","url":null,"abstract":"Abstract Let G be a 10-regular graph which does not contain any 4-cycles. In this paper, we prove that G can be decomposed into paths of length 5, such that every vertex is a terminal of exactly two paths.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589257","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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