Let G = ( V , E , F ) be a connected, loopless, and bridgeless plane graph, with vertex set V , e... more Let G = ( V , E , F ) be a connected, loopless, and bridgeless plane graph, with vertex set V , edge set E , and face set F . For X ? { V , E , F , V ? E , V ? F , E ? F , V ? E ? F } , two elements x and y of X are facially adjacent in G if they are incident, or they are adjacent vertices, or adjacent faces, or facially adjacent edges (i.e.?edges that are consecutive on the boundary walk of a face of G ). A k -colouring is facial with respect to X if there is a k -colouring of elements of X such that facially adjacent elements of X receive different colours. We prove that: (i) Every plane graph G = ( V , E , F ) has a facial 8-colouring with respect to X = V ? E ? F (i.e.?a facial entire 8-colouring). Moreover, there is plane graph requiring at least 7 colours in any such colouring. (ii) Every plane graph G = ( V , E , F ) has a facial 6-colouring with respect to X = E ? F , in other words, a facial edge-face 6-colouring.
Abstract Let G be a plane graph. A facial path of G is any path which is a consecutive part of th... more Abstract Let G be a plane graph. A facial path of G is any path which is a consecutive part of the boundary walk of a face of G . Two edges e 1 and e 2 of G are facially adjacent if they are consecutive on a facial path of G . Two edges e 1 and e 3 are facially semi-adjacent if they are not facially adjacent and there is a third edge e 2 which is facially adjacent with both e 1 and e 3 , and the edges e 1 , e 2 , e 3 are consecutive (in this order) on a facial path. An edge-labeling of G with labels 1 , 2 , … , k is a facial L ( 2 , 1 ) -edge-labeling if facially adjacent edges have labels which differ by at least 2 and facially semi-adjacent edges have labels which differ by at least 1. The minimum k for which a plane graph admits a facial L ( 2 , 1 ) -edge-labeling is called the facial L ( 2 , 1 ) -edge-labeling index. In this paper, we prove that the facial L ( 2 , 1 ) -edge-labeling index of any tree T is at most 7; moreover, this bound is tight. In the case when T has no vertex of degree 3 the upper bound for this parameter is 6, which is also tight. If T is without vertices of degree 2 and 3, then its facial L ( 2 , 1 ) -edge-labeling index is at most 5; moreover, this bound is also tight. Finally, we characterize all trees having facial L ( 2 , 1 ) -edge-labeling index exactly 4.
In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with mi... more In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with minimum degree at least 2 and a given girth g . A 3-path of type ( i , j , k ) is a path u v w on three vertices u , v , and w such that the degree of u (resp. v , resp. w ) is at most i (resp. j , resp. k ). The elements i , j , k are called parameters of the type. The set S of types of paths is unavoidable for a family F of graphs if each graph G from F contains a path of the type from S . An unavoidable set S of types of paths is optimal for the family F if neither any type can be omitted from S , nor any parameter of any type from S can be decreased.We prove that the set S g (resp. S ' g ) is an optimal set of types of 3-paths for the family of plane graphs having ? ( G ) ? 2 and girth g ( G ) ? g where (i) S 5 = { ( 2 , ∞ , 2 ) , ( 2 , 3 , 5 ) , ( 2 , 4 , 3 ) , ( 3 , 3 , 3 ) } ,(ii) S 7 = { ( 2 , 3 , 3 ) , ( 2 , 5 , 2 ) } , S 7 ' = { ( 2 , 2 , 6 ) , ( 2 , 3 , 3 ) , ( 2 , 4 , 2 ) } ,(iii) S 8 = { ( 2 , 2 , 5 ) , ( 2 , 3 , 2 ) } ,(iv) S 10 = { ( 2 , 4 , 2 ) } , S 10 ' = { ( 2 , 2 , 3 ) , ( 2 , 3 , 2 ) } ,(v) S 11 = { ( 2 , 2 , 3 ) } .
Electronic Notes in Discrete Mathematics, Jul 1, 2017
Abstract In this paper we give a survey on several types of colourings of elements of graphs by d... more Abstract In this paper we give a survey on several types of colourings of elements of graphs by different types of labellings.
Let G = ( V , E , F ) be a connected, loopless, and bridgeless plane graph, with vertex set V , e... more Let G = ( V , E , F ) be a connected, loopless, and bridgeless plane graph, with vertex set V , edge set E , and face set F . For X ? { V , E , F , V ? E , V ? F , E ? F , V ? E ? F } , two elements x and y of X are facially adjacent in G if they are incident, or they are adjacent vertices, or adjacent faces, or facially adjacent edges (i.e.?edges that are consecutive on the boundary walk of a face of G ). A k -colouring is facial with respect to X if there is a k -colouring of elements of X such that facially adjacent elements of X receive different colours. We prove that: (i) Every plane graph G = ( V , E , F ) has a facial 8-colouring with respect to X = V ? E ? F (i.e.?a facial entire 8-colouring). Moreover, there is plane graph requiring at least 7 colours in any such colouring. (ii) Every plane graph G = ( V , E , F ) has a facial 6-colouring with respect to X = E ? F , in other words, a facial edge-face 6-colouring.
Abstract Let G be a plane graph. A facial path of G is any path which is a consecutive part of th... more Abstract Let G be a plane graph. A facial path of G is any path which is a consecutive part of the boundary walk of a face of G . Two edges e 1 and e 2 of G are facially adjacent if they are consecutive on a facial path of G . Two edges e 1 and e 3 are facially semi-adjacent if they are not facially adjacent and there is a third edge e 2 which is facially adjacent with both e 1 and e 3 , and the edges e 1 , e 2 , e 3 are consecutive (in this order) on a facial path. An edge-labeling of G with labels 1 , 2 , … , k is a facial L ( 2 , 1 ) -edge-labeling if facially adjacent edges have labels which differ by at least 2 and facially semi-adjacent edges have labels which differ by at least 1. The minimum k for which a plane graph admits a facial L ( 2 , 1 ) -edge-labeling is called the facial L ( 2 , 1 ) -edge-labeling index. In this paper, we prove that the facial L ( 2 , 1 ) -edge-labeling index of any tree T is at most 7; moreover, this bound is tight. In the case when T has no vertex of degree 3 the upper bound for this parameter is 6, which is also tight. If T is without vertices of degree 2 and 3, then its facial L ( 2 , 1 ) -edge-labeling index is at most 5; moreover, this bound is also tight. Finally, we characterize all trees having facial L ( 2 , 1 ) -edge-labeling index exactly 4.
In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with mi... more In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with minimum degree at least 2 and a given girth g . A 3-path of type ( i , j , k ) is a path u v w on three vertices u , v , and w such that the degree of u (resp. v , resp. w ) is at most i (resp. j , resp. k ). The elements i , j , k are called parameters of the type. The set S of types of paths is unavoidable for a family F of graphs if each graph G from F contains a path of the type from S . An unavoidable set S of types of paths is optimal for the family F if neither any type can be omitted from S , nor any parameter of any type from S can be decreased.We prove that the set S g (resp. S ' g ) is an optimal set of types of 3-paths for the family of plane graphs having ? ( G ) ? 2 and girth g ( G ) ? g where (i) S 5 = { ( 2 , ∞ , 2 ) , ( 2 , 3 , 5 ) , ( 2 , 4 , 3 ) , ( 3 , 3 , 3 ) } ,(ii) S 7 = { ( 2 , 3 , 3 ) , ( 2 , 5 , 2 ) } , S 7 ' = { ( 2 , 2 , 6 ) , ( 2 , 3 , 3 ) , ( 2 , 4 , 2 ) } ,(iii) S 8 = { ( 2 , 2 , 5 ) , ( 2 , 3 , 2 ) } ,(iv) S 10 = { ( 2 , 4 , 2 ) } , S 10 ' = { ( 2 , 2 , 3 ) , ( 2 , 3 , 2 ) } ,(v) S 11 = { ( 2 , 2 , 3 ) } .
Electronic Notes in Discrete Mathematics, Jul 1, 2017
Abstract In this paper we give a survey on several types of colourings of elements of graphs by d... more Abstract In this paper we give a survey on several types of colourings of elements of graphs by different types of labellings.
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