On Pathos Semitotal and Total Block Graph of a Tree

The concept of pathos of a graph G was introduced by Harary [2], as a collection of minimum number of line disjoint open paths whose union is G. The path number of a graph G is the number of paths in pathos. A new concept of a graph valued functions called the semitotal and total block graph of a graph was introduced by Kulli

The four color theorem: from graph theory to proof assistants.

openLa tesi inizialmente descrive i fondamenti della teoria dei grafi con le principali nozioni per affrontare il teorema dei sei, cinque e infine dei quattro colori. Quest'ultimo viene descritto dal punto di vista storico e viene fornita una traccia della dimostrazione, per poi indagare gli aspetti legati all'utilizzo di proof assistant.First, it describes the basic notions of graph theory in order to face the six, five and finally the four color theorem. This last problem is treated from an historical point of view and the main steps of the proof are given. Finally, some aspects linked to proof assistants are examine

Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine

International Journal of Mathematical Combinatorics, Vol.6A

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Transversal Problems In Sparse Graphs

Graph transversals are a classical branch of graph algorithms. In such a problem, one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family $\mathcal F$. In the first portion of this thesis we show two results related to even cycle transversal. %%Note rephrase this later. In Chapter \ref{ECTChapter}, we present our 47/7-approximation for even cycle transversal. To do this, we first apply a graph ``compression" method of Fiorini et al. % \cite{FioriniJP2010} which we describe in Chapter \ref{PreliminariesChapter}. For the analysis we repurpose the theory behind the 18/7-approximation for ``uncrossable" feedback vertex set problems of Berman and Yaroslavtsev %% \cite{BermanY2012} noting that we do not need the larger ``witness" cycles to be a cycle that we need to hit. This we do in Chapter \ref{BermanYaroChapter}. In Chapter \ref{ErdosPosaChapter} we present a simple proof of an Erdos Posa result, that for any natural number $k$ a planar graph $G$ either contains $k$ vertex disjoint even cycles, or a set $X$ of at most $9k$ such that $G \backslash X$ contains no even cycle. In the rest of this thesis, we show a result for dominating set. A dominating set $S$ in a graph is a set of vertices such that each node is in $S$ or adjacent to $S$. In Chapter 6 we present a primal-dual $(a+1)$-approximation for minimum weight dominating set in graphs of arboricity $a$. At the end, we propose five open problems for future research