We introduce and study the dominated edge coloring of a graph. A graph \(G\), is proper \(G\) such that each color class by at least one \(G\). The minimum number colors among all called chromatic number, denoted \(\chi_{dom}^{\prime}(G)\). obtain some properties \(\chi_{dom}^{\prime}(G)\) compute it for specific graphs. Also examine effects on \(\chi_{dom}^{\prime}(G)\), when modified operatio...

The graph sharing game is played by two players, Alice and Bob, on a connected graph G with non-negative weights assigned to the vertices. Starting with Alice, the players take the vertices of G one by one, in each move keeping the set of all taken vertices connected, until the whole G has been taken. Each player wants to maximize the total weight of the vertices they have gathered. It is prove...

In this paper we continue our investigations from [HM01] regarding spanning subgraphs which imply the existence of cycle double covers. We prove that if a cubic graph G has a spanning subgraph isomorphic to a subdivision of a bridgeless cubic graph on at most 10 vertices then G has a CDC. A notable result is thus that a cubic graph with a spanning Petersen minor has a CDC, a result also obtaine...

Raspaud, A., Postman tours and cycle covers, Discrete Mathematics 111 (1993) 447-454. Let G be a bridgeless graph. We show that the length of a shortest postman tour is at most IF(G)1 + 1 k’(G)1 3 and that, if G is a minimally 2-edge connected graph, then the length is at most 21 V(G)l-2. We then deduce results concerning the length of a shortest cycle cover for graphs containing no subdivision...

Cun-Quan Zhang DEPARTMENT OF MATHEMA TICS WEST VIRGINIA UNIVERSITY MORGANTOWN, WEST VIRGINIA A (1,2)-eulerian weight w of a graph is hamiltonian if every faithful cover of w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no subdivision of the Petersen graph. We prove that if G admits a hamiltonian weight then G is uniquely 3-edge-colorable. © 1995 John Wiley & ...

Let G be a graph and let S(G), M(G), T(G) the subdivision, middle, total of G, respectively. dim(G), edim(G), mdim(G) metric dimension, edge mixed dimension In this paper, for subdivision it is proved that max{dim(G), edim(G)} ≤ mdim(S(G)) mdim(G). A family graphs Gn constructed which mdim(Gn) − mdim(S(Gn)) ≥ 2 holds shows inequality can strict, while cactus = For middle dim(M(G)) holds, if tre...

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K5. In this paper, we prove this conjecture for graphs containing K− 4 . AMS Subject Classification: 05C

We study the complexity and the I/O-efficient computation of flow on triangulated terrains. We present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/o-efficiently. We use the descent graph to obtain i/o-efficient algorithms for computing river networks and watershed-area maps in O(Sort(d + r)) i/o’s, where r is the complexity of the river network a...