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    Wyświetlanie 1-3 z 3
    Tytuł:
    Characterizations of Graphs Having Large Proper Connection Numbers
    Autorzy:
    Lumduanhom, Chira
    Laforge, Elliot
    Zhang, Ping
    Powiązania:
    https://bibliotekanauki.pl/articles/31340917.pdf
    Data publikacji:
    2016-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge coloring
    proper-path coloring
    strong proper-path coloring
    Opis:
    Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u− v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1,m − 2 or m − 3.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2016, 36, 2; 439-453
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Eulerian irregularity in graphs
    Autorzy:
    Andrews, Eric
    Lumduanhom, Chira
    Zhang, Ping
    Powiązania:
    https://bibliotekanauki.pl/articles/31232740.pdf
    Data publikacji:
    2014-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Eulerian walks
    Eulerian irregularity
    Opis:
    A closed walk in a connected graph $G$ that contains every edge of $G$ exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph $G$ is Eulerian if and only if every vertex of $G$ is even. An Eulerian walk in a connected graph $G$ is a closed walk that contains every edge of $G$ at least once, while an irregular Eulerian walk in $G$ is an Eulerian walk that encounters no two edges of $G$ the same number of times. The minimum length of an irregular Eulerian walk in $G$ is called the Eulerian irregularity of $G$ and is denoted by $EI(G)$. It is known that if $G$ is a nontrivial connected graph of size $m$, then \(\binom{m+1}{2} \le EI(G) \le 2 \binom{m+1}{2}\). A necessary and sufficient condition has been established for all pairs $k, m$ of positive integers for which there is a nontrivial connected graph $G$ of size $m$ with $EI(G)=k$. A subgraph $F$ in a graph $G$ is an even subgraph of $G$ if every vertex of $F$ is even. We present a formula for the Eulerian irregularity of a graph in terms of the size of certain even subgraph of the graph. Furthermore, Eulerian irregularities are determined for all graphs of cycle rank 2 and all complete bipartite graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 2; 391-408
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Monochromatic Subgraphs of Edge-Colored Complete Graphs
    Autorzy:
    Andrews, Eric
    Fujie, Futaba
    Kolasinski, Kyle
    Lumduanhom, Chira
    Yusko, Adam
    Powiązania:
    https://bibliotekanauki.pl/articles/30147209.pdf
    Data publikacji:
    2014-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Ramsey number
    monochromatic Ramsey number
    common monochromatic subgraph
    maximal common monochromatic subgraph
    Opis:
    In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the monochromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey number of G and H is bounded below by mr(G,H). The monochromatic Ramsey numbers of graphs belonging to some common classes of graphs are studied. We also investigate another concept closely related to the standard Ramsey numbers and monochromatic Ramsey numbers of graphs. For a fixed integer n ≥ 3, consider a nonempty subgraph G of order at most n containing no isolated vertices. Then G is a common monochromatic subgraph of Kn if every red-blue coloring of Kn results in a monochromatic copy of G. Furthermore, G is a maximal common monochromatic subgraph of Kn if G is a common monochromatic subgraph of Kn that is not a proper subgraph of any common monochromatic subgraph of Kn. Let S(n) and S*(n) be the sets of common monochromatic subgraphs and maximal common monochromatic subgraphs of Kn, respectively. Thus, G ∈ S(n) if and only if R(G,G) = mr(G,G) ≤ n. We determine the sets S(n) and S*(n) for 3 ≤ n ≤ 8.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 1; 5-22
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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