{"title":"Walk Domination and HHD-Free Graphs","authors":"Silvia B. Tondato","doi":"10.1007/s00373-024-02771-y","DOIUrl":"https://doi.org/10.1007/s00373-024-02771-y","url":null,"abstract":"<p>HHD-free is the class of graphs which contain no house, hole, or domino as induced subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and via <span>(m^3)</span>-convexity. In this paper we present new characterizations of HHD-free via domination of paths and walks. To achieve this, in particular we concentrate our attention on <span>(m_3)</span> path, i.e, an induced path of length at least 3 between two non-adjacent vertices in a graph <i>G</i>. We show that the domination between induced paths, paths and walks versus <span>(m_3)</span> paths, gives rise to characterization of HHD-free. We also characterize the class of graphs in which every <span>(m_3)</span> path dominates every path, induced path, walk, and <span>(m_3)</span> path, respectively.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"92 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578553","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":"Using Euler’s Formula to Find the Lower Bound of the Page Number","authors":"Bin Zhao, Peng Li, Jixiang Meng, Yuepeng Zhang","doi":"10.1007/s00373-024-02775-8","DOIUrl":"https://doi.org/10.1007/s00373-024-02775-8","url":null,"abstract":"<p>The concept of book embedding, originating in computer science, has found extensive applications in various problem domains. A book embedding of a graph <i>G</i> involves arranging the vertices of <i>G</i> in an order along a line and assigning the edges to one or more half-planes. The page number of a graph is the smallest possible number of half-planes for any book embedding of the graph. Determining the page number is a key aspect of book embedding and carries significant importance. This paper aims to investigate the non-trivial lower bound of the page number for both a graph <i>G</i> and a random graph <span>(Gin mathcal {G}(n,p))</span> by incorporating two seemingly unrelated concepts: edge-arboricity and Euler’s Formula. Our analysis reveals that for a graph <i>G</i>, which is not a path, <span>(pn(G)ge lceil frac{1}{3} a_1(G)rceil )</span>, where <span>(a_1(G))</span> denotes the edge-arboricity of <i>G</i>, and for an outerplanar graph, the lower bound is optimal. For <span>(Gin mathcal {G}(n,p))</span>, <span>(pn(G)ge lceil frac{1}{6}np(1-o(1))rceil )</span> with high probability, as long as <span>(frac{c}{n}le ple frac{root 2 of {3(n-1)}}{nlog {n}})</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578832","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":"Path Saturation Game on Six Vertices","authors":"Paul Balister, Ali Dogan","doi":"10.1007/s00373-024-02767-8","DOIUrl":"https://doi.org/10.1007/s00373-024-02767-8","url":null,"abstract":"<p>Given a family <span>(mathcal {F})</span> of graphs, we say that a graph <i>G</i> is <span>(mathcal {F})</span>-saturated if <i>G</i> does not contain any member of <span>(mathcal {F})</span>, but for any edge <span>(ein E(overline{G}))</span> the graph <span>(G+e)</span> does contain a member of <span>(mathcal {F})</span>. The <span>(mathcal {F})</span>-<i>saturation game</i> is played by two players starting with an empty graph and adding an edge on their turn without making a member of <span>(mathcal {F})</span>. The game ends when the graph is <span>(mathcal {F})</span>-saturated. One of the players wants to maximize the number edges in the final graph, while the other wants to minimize it. The <i>game saturation number</i> is the number of edges in the final graph given the optimal play by both players. In the present paper we study <span>(mathcal {F})</span>-saturation game when <span>(mathcal {F}={P_6})</span> consists of the single path on 6 vertices.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578884","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":"Some Results on the Rainbow Vertex-Disconnection Colorings of Graphs","authors":"Yindi Weng","doi":"10.1007/s00373-024-02762-z","DOIUrl":"https://doi.org/10.1007/s00373-024-02762-z","url":null,"abstract":"<p>Let <i>G</i> be a nontrivial connected and vertex-colored graph. A vertex subset <i>X</i> is called <i>rainbow</i> if any two vertices in <i>X</i> have distinct colors. The graph <i>G</i> is called <i>rainbow vertex-disconnected</i> if for any two vertices <i>x</i> and <i>y</i> of <i>G</i>, there exists a vertex subset <i>S</i> such that when <i>x</i> and <i>y</i> are nonadjacent, <i>S</i> is rainbow and <i>x</i> and <i>y</i> belong to different components of <span>(G-S)</span>; whereas when <i>x</i> and <i>y</i> are adjacent, <span>(S+x)</span> or <span>(S+y)</span> is rainbow and <i>x</i> and <i>y</i> belong to different components of <span>((G-xy)-S)</span>. For a connected graph <i>G</i>, the <i>rainbow vertex-disconnection number</i> of <i>G</i>, <i>rvd</i>(<i>G</i>), is the minimum number of colors that are needed to make <i>G</i> rainbow vertex-disconnected. In this paper, we prove for any <span>(K_4)</span>-minor free graph, <span>(rvd(G)le Delta (G))</span> and the bound is sharp. We show it is <i>NP</i>-complete to determine the rainbow vertex-disconnection numbers for bipartite graphs and split graphs. Moreover, we show for every <span>(epsilon >0)</span>, it is impossible to efficiently approximate the rainbow vertex-disconnection number of any bipartite graph and split graph within a factor of <span>(n^{frac{1}{3}-epsilon })</span> unless <span>(ZPP=NP)</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"136 3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578558","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":"Removable Edges in Claw-Free Bricks","authors":"","doi":"10.1007/s00373-024-02769-6","DOIUrl":"https://doi.org/10.1007/s00373-024-02769-6","url":null,"abstract":"<h3>Abstract</h3> <p>An edge <em>e</em> in a matching covered graph <em>G</em> is <em>removable</em> if <span> <span>(G-e)</span> </span> is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A <em>brick</em> is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than <span> <span>(K_4)</span> </span> and <span> <span>(overline{C_6})</span> </span> has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick <em>G</em> with more than 6 vertices has at least 5|<em>V</em>(<em>G</em>)|/8 removable edges.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"138 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578616","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":"Strongly Regular Graphs from Pseudocyclic Association Schemes","authors":"Koji Momihara, Sho Suda","doi":"10.1007/s00373-024-02764-x","DOIUrl":"https://doi.org/10.1007/s00373-024-02764-x","url":null,"abstract":"<p>In this paper, we give a construction of strongly regular graphs from pseudocyclic association schemes, which is a common generalization of two constructions given by Fujisaki (2004). Furthermore, we prove that the pseudocyclic association scheme arising from the action of PGL(2, <i>q</i>) to the set of exterior lines in PG(2, <i>q</i>), called the elliptic scheme, under the assumption that <span>(q=2^m)</span> with <i>m</i> an odd prime satisfies the condition of our new construction. As a consequence, we obtain a new infinite family of strongly regular graphs of Latin square type with non-prime-power number of vertices.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297881","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":"Neighborhood Balanced Colorings of Graphs","authors":"Bryan Freyberg, Alison Marr","doi":"10.1007/s00373-024-02766-9","DOIUrl":"https://doi.org/10.1007/s00373-024-02766-9","url":null,"abstract":"<p>Given a simple graph <i>G</i>, we ask when <i>V</i>(<i>G</i>) may be partitioned into two sets such that every vertex has an equal number of neighbors from each set. We establish a number of results for common families of graphs and completely classify 4-regular circulants which posses this property.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"233 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297678","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":"Extremal Edge General Position Sets in Some Graphs","authors":"","doi":"10.1007/s00373-024-02770-z","DOIUrl":"https://doi.org/10.1007/s00373-024-02770-z","url":null,"abstract":"<h3>Abstract</h3> <p>A set of edges <span> <span>(Xsubseteq E(G))</span> </span> of a graph <em>G</em> is an edge general position set if no three edges from <em>X</em> lie on a common shortest path. The edge general position number <span> <span>({textrm{gp}}_{textrm{e}}(G))</span> </span> of <em>G</em> is the cardinality of a largest edge general position set in <em>G</em>. Graphs <em>G</em> with <span> <span>({textrm{gp}}_{{textrm{e}}}(G) = |E(G)| - 1)</span> </span> and with <span> <span>({textrm{gp}}_{{textrm{e}}}(G) = 3)</span> </span> are respectively characterized. Sharp upper and lower bounds on <span> <span>({textrm{gp}}_{{textrm{e}}}(G))</span> </span> are proved for block graphs <em>G</em> and exact values are determined for several specific block graphs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"56 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297708","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 Non-degenerate Berge–Turán Problems","authors":"Dániel Gerbner","doi":"10.1007/s00373-024-02757-w","DOIUrl":"https://doi.org/10.1007/s00373-024-02757-w","url":null,"abstract":"<p>Given a hypergraph <span>({{mathcal {H}}})</span> and a graph <i>G</i>, we say that <span>({{mathcal {H}}})</span> is a <i>Berge</i>-<i>G</i> if there is a bijection between the hyperedges of <span>({{mathcal {H}}})</span> and the edges of <i>G</i> such that each hyperedge contains its image. We denote by <span>(textrm{ex}_k(n,Berge- F))</span> the largest number of hyperedges in a <i>k</i>-uniform Berge-<i>F</i>-free graph. Let <span>(textrm{ex}(n,H,F))</span> denote the largest number of copies of <i>H</i> in <i>n</i>-vertex <i>F</i>-free graphs. It is known that <span>(textrm{ex}(n,K_k,F)le textrm{ex}_k(n,Berge- F)le textrm{ex}(n,K_k,F)+textrm{ex}(n,F))</span>, thus if <span>(chi (F)>r)</span>, then <span>(textrm{ex}_k(n,Berge- F)=(1+o(1)) textrm{ex}(n,K_k,F))</span>. We conjecture that <span>(textrm{ex}_k(n,Berge- F)=textrm{ex}(n,K_k,F))</span> in this case. We prove this conjecture in several instances, including the cases <span>(k=3)</span> and <span>(k=4)</span>. We prove the general bound <span>(textrm{ex}_k(n,Berge- F)= textrm{ex}(n,K_k,F)+O(1))</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297677","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":"Cycle Isolation of Graphs with Small Girth","authors":"Gang Zhang, Baoyindureng Wu","doi":"10.1007/s00373-024-02768-7","DOIUrl":"https://doi.org/10.1007/s00373-024-02768-7","url":null,"abstract":"<p>Let <i>G</i> be a graph. A subset <span>(D subseteq V(G))</span> is a decycling set of <i>G</i> if <span>(G-D)</span> contains no cycle. A subset <span>(D subseteq V(G))</span> is a cycle isolating set of <i>G</i> if <span>(G-N[D])</span> contains no cycle. The decycling number and cycle isolation number of <i>G</i>, denoted by <span>(phi (G))</span> and <span>(iota _c(G))</span>, are the minimum cardinalities of a decycling set and a cycle isolating set of <i>G</i>, respectively. Dross, Montassier and Pinlou (Discrete Appl Math 214:99–107, 2016) conjectured that if <i>G</i> is a planar graph of size <i>m</i> and girth at least <i>g</i>, then <span>(phi (G) le frac{m}{g})</span>. So far, this conjecture remains open. Recently, the authors proposed an analogous conjecture that if <i>G</i> is a connected graph of size <i>m</i> and girth at least <i>g</i> that is different from <span>(C_g)</span>, then <span>(iota _c(G) le frac{m+1}{g+2})</span>, and they presented a proof for the initial case <span>(g=3)</span>. In this paper, we further prove that for the cases of girth at least 4, 5 and 6, this conjecture is true. The extremal graphs of results above are characterized.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"22 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297813","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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