Displaying similar documents to “Saturation numbers for linear forests $P_6 + tP_2$”

Generalized 3-edge-connectivity of Cartesian product graphs

Yuefang Sun (2015)

Czechoslovak Mathematical Journal

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The generalized k -connectivity κ k ( G ) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k -edge-connectivity which is defined as λ k ( G ) = min { λ ( S ) : S V ( G ) and | S | = k } , where λ ( S ) denotes the maximum number of pairwise edge-disjoint trees T 1 , T 2 , ... , T in G such that S V ( T i ) for 1 i . In this paper we prove that for any two connected graphs G and H we have λ 3 ( G H ) λ 3 ( G ) + λ 3 ( H ) , where G H is the Cartesian product of G and H . Moreover, the bound is sharp. We also...

Even factor of bridgeless graphs containing two specified edges

Nastaran Haghparast, Dariush Kiani (2018)

Czechoslovak Mathematical Journal

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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let G be a bridgeless simple graph with minimum degree at least 3 . Jackson and Yoshimoto (2007) showed that G has an even factor containing two arbitrary prescribed edges. They also proved that G has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges e 1 and e 2 of G , there is an even factor containing e 1 and e 2 ...

Matchings in complete bipartite graphs and the r -Lah numbers

Gábor Nyul, Gabriella Rácz (2021)

Czechoslovak Mathematical Journal

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We give a graph theoretic interpretation of r -Lah numbers, namely, we show that the r -Lah number n k r counting the number of r -partitions of an ( n + r ) -element set into k + r ordered blocks is just equal to the number of matchings consisting of n - k edges in the complete bipartite graph with partite sets of cardinality n and n + 2 r - 1 ( 0 k n , r 1 ). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for r -Stirling numbers of the second kind. ...

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph G , denoted by L ( G ) , has E ( G ) as its vertex set, where two vertices in L ( G ) are adjacent if and only if the corresponding edges in G have a vertex in common. For a graph H , define σ ¯ 2 ( H ) = min { d ( u ) + d ( v ) : u v E ( H ) } . Let H be a 2-connected claw-free simple graph of order n with δ ( H ) 3 . We show that, if σ ¯ 2 ( H ) 1 7 ( 2 n - 5 ) and n is sufficiently large, then either H is traceable or the Ryjáček’s closure cl ( H ) = L ( G ) , where G is an essentially 2 -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

The Turán number of the graph 3 P 4

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let e x ( n , G ) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let P i denote a path consisting of i vertices and let m P i denote m disjoint copies of P i . In this paper we count e x ( n , 3 P 4 ) .

The small Ree group 2 G 2 ( 3 2 n + 1 ) and related graph

Alireza K. Asboei, Seyed S. S. Amiri (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group. The main supergraph 𝒮 ( G ) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o ( x ) o ( y ) or o ( y ) o ( x ) . In this paper, we will show that G 2 G 2 ( 3 2 n + 1 ) if and only if 𝒮 ( G ) 𝒮 ( 2 G 2 ( 3 2 n + 1 ) ) . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group 2 G 2 ( 3 2 n + 1 ) .

On γ-labelings of trees

Gary Chartrand, David Erwin, Donald W. VanderJagt, Ping Zhang (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is v a l ( f ) = Σ e E ( G ) f ' K ( e ) . The maximum value of a γ-labeling of G is defined as v a l m a x ( G ) = m a x v a l ( f ) : f i s a γ - l a b e l i n g o f G ; while the minimum value of a γ-labeling of G is v a l m i n ( G ) = m i n v a l ( f ) : f i s a γ - l a b e l i n g o f G ; The values v a l m a x ( S p , q ) and v a l m i n ( S p , q ) are determined for double stars S p , q . We present characterizations of connected graphs G of order n for which...

Note on improper coloring of 1 -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

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A graph G = ( V , E ) is called improperly ( d 1 , , d k ) -colorable if the vertex set V can be partitioned into subsets V 1 , , V k such that the graph G [ V i ] induced by the vertices of V i has maximum degree at most d i for all 1 i k . In this paper, we mainly study the improper coloring of 1 -planar graphs and show that 1 -planar graphs with girth at least 7 are ( 2 , 0 , 0 , 0 ) -colorable.

On g c -colorings of nearly bipartite graphs

Yuzhuo Zhang, Xia Zhang (2018)

Czechoslovak Mathematical Journal

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Let G be a simple graph, let d ( v ) denote the degree of a vertex v and let g be a nonnegative integer function on V ( G ) with 0 g ( v ) d ( v ) for each vertex v V ( G ) . A g c -coloring of G is an edge coloring such that for each vertex v V ( G ) and each color c , there are at least g ( v ) edges colored c incident with v . The g c -chromatic index of G , denoted by χ g c ' ( G ) , is the maximum number of colors such that a g c -coloring of G exists. Any simple graph G has the g c -chromatic index equal to δ g ( G ) or δ g ( G ) - 1 , where δ g ( G ) = min v V ( G ) d ( v ) / g ( v ) . A graph G is nearly bipartite,...

The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

Halina Bielak, Kinga Dąbrowska (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The Ramsey number R ( G , H ) for a pair of graphs G and H is defined as the smallest integer n such that, for any graph F on n vertices, either F contains G or F ¯ contains H as a subgraph, where F ¯ denotes the complement of F . We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers R ( K 1 + L n , P m ) and R ( K 1 + L n , C m ) for some integers m , n , where L n is...

Maximum bipartite subgraphs in H -free graphs

Jing Lin (2022)

Czechoslovak Mathematical Journal

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Given a graph G , let f ( G ) denote the maximum number of edges in a bipartite subgraph of G . Given a fixed graph H and a positive integer m , let f ( m , H ) denote the minimum possible cardinality of f ( G ) , as G ranges over all graphs on m edges that contain no copy of H . In this paper we prove that f ( m , θ k , s ) 1 2 m + Ω ( m ( 2 k + 1 ) / ( 2 k + 2 ) ) , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write K k ' and K t , s ' for the subdivisions of K k and K t , s . We show that f ( m , K k ' ) 1 2 m + Ω ( m ( 5 k - 8 ) / ( 6 k - 10 ) ) and f ( m , K t , s ' ) 1 2 m + Ω ( m ( 5 t - 1 ) / ( 6 t - 2 ) ) , improving a result of Q. Zeng, J. Hou. We also give lower bounds on...

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let G 1 and G 2 be two given graphs. The Ramsey number R ( G 1 , G 2 ) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G ¯ contains a G 2 . Parsons gave a recursive formula to determine the values of R ( P n , K 1 , m ) , where P n is a path on n vertices and K 1 , m is a star on m + 1 vertices. In this note, we study the Ramsey numbers R ( P n , K 1 F m ) , where F m is a linear forest on m vertices. We determine the exact values of R ( P n , K 1 F m ) for the cases m n and m 2 n , and for the case that F m has no odd component. Moreover, we...

A note on the double Roman domination number of graphs

Xue-Gang Chen (2020)

Czechoslovak Mathematical Journal

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For a graph G = ( V , E ) , a double Roman dominating function is a function f : V { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then the vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then the vertex v must have at least one neighbor with f ( w ) 2 . The weight of a double Roman dominating function f is the sum f ( V ) = v V f ( v ) . The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γ dR ( G ) . In this paper, we establish a new...

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

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Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G , a hereditary graph property 𝒫 and l 1 we define χ 𝒫 , l ' ( G ) to be the minimum number of colours needed to properly colour the edges of G , such that any subgraph of G induced by edges coloured by (at most) l colours is in 𝒫 . We present a necessary and sufficient condition for the existence of χ 𝒫 , l ' ( G ) . We focus on edge-colourings of graphs with respect to the hereditary...

The potential-Ramsey number of K n and K t - k

Jin-Zhi Du, Jian Hua Yin (2022)

Czechoslovak Mathematical Journal

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A nonincreasing sequence π = ( d 1 , ... , d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π . Given two graphs G 1 and G 2 , A. Busch et al. (2014) introduced the potential-Ramsey number of G 1 and G 2 , denoted by r pot ( G 1 , G 2 ) , as the smallest nonnegative integer m such that for every m -term graphic sequence π , there is a realization G of π with G 1 G or with G 2 G ¯ , where G ¯ is the complement of G . For t 2 and 0 k t 2 , let K t - k be the graph...